statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
mk_eq (x : pSet) : @eq Set ⟦x⟧ (mk x) | rfl | theorem | Set.mk_eq | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_out : ∀ x : Set, mk x.out = x | quotient.out_eq | theorem | Set.mk_out | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"quotient.out_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq {x y : pSet} : mk x = mk y ↔ equiv x y | quotient.eq | theorem | Set.eq | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"pSet",
"quotient.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sound {x y : pSet} (h : pSet.equiv x y) : mk x = mk y | quotient.sound h | theorem | Set.sound | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet",
"pSet.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact {x y : pSet} : mk x = mk y → pSet.equiv x y | quotient.exact | theorem | Set.exact | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet",
"pSet.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_mk {n f x} :
(@resp.eval (n+1) f : Set → arity Set n) (mk x) = resp.eval n (resp.f f x) | rfl | lemma | Set.eval_mk | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"arity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem : Set → Set → Prop | quotient.lift₂ pSet.mem
(λ x y x' y' hx hy, propext ((mem.congr_left hx).trans (mem.congr_right hy))) | def | Set.mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"pSet.mem"
] | The membership relation for ZFC sets is inherited from the membership relation for pre-sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_mem_iff {x y : pSet} : mk x ∈ mk y ↔ x ∈ y | iff.rfl | theorem | Set.mk_mem_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set (u : Set.{u}) : set Set.{u} | {x | x ∈ u} | def | Set.to_set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Convert a ZFC set into a `set` of ZFC sets | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_to_set (a u : Set.{u}) : a ∈ u.to_set ↔ a ∈ u | iff.rfl | theorem | Set.mem_to_set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_to_set (x : Set.{u}) : small.{u} x.to_set | quotient.induction_on x $ λ a, begin
let f : a.type → (mk a).to_set := λ i, ⟨mk $ a.func i, func_mem a i⟩,
suffices : function.surjective f,
{ exact small_of_surjective this },
rintro ⟨y, hb⟩,
induction y using quotient.induction_on,
cases hb with i h,
exact ⟨i, subtype.coe_injective (quotient.sound h.sym... | instance | Set.small_to_set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"small_of_surjective",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty (u : Set) : Prop | u.to_set.nonempty | def | Set.nonempty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | A nonempty set is one that contains some element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_def (u : Set) : u.nonempty ↔ ∃ x, x ∈ u | iff.rfl | theorem | Set.nonempty_def | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_of_mem {x u : Set} (h : x ∈ u) : u.nonempty | ⟨x, h⟩ | theorem | Set.nonempty_of_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_to_set_iff {u : Set} : u.to_set.nonempty ↔ u.nonempty | iff.rfl | theorem | Set.nonempty_to_set_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset (x y : Set.{u}) | ∀ ⦃z⦄, z ∈ x → z ∈ y | def | Set.subset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | `x ⊆ y` as ZFC sets means that all members of `x` are members of `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_subset : has_subset Set | ⟨Set.subset⟩ | instance | Set.has_subset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_def {x y : Set.{u}} : x ⊆ y ↔ ∀ ⦃z⦄, z ∈ x → z ∈ y | iff.rfl | lemma | Set.subset_def | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_iff : Π {x y : pSet}, mk x ⊆ mk y ↔ x ⊆ y | | ⟨α, A⟩ ⟨β, B⟩ := ⟨λ h a, @h ⟦A a⟧ (mem.mk A a),
λ h z, quotient.induction_on z (λ z ⟨a, za⟩, let ⟨b, ab⟩ := h a in ⟨b, za.trans ab⟩)⟩ | theorem | Set.subset_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_subset_iff {x y : Set} : x.to_set ⊆ y.to_set ↔ x ⊆ y | by simp [subset_def, set.subset_def] | theorem | Set.to_set_subset_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"set.subset_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {x y : Set.{u}} : (∀ z : Set.{u}, z ∈ x ↔ z ∈ y) → x = y | quotient.induction_on₂ x y (λ u v h, quotient.sound (mem.ext (λ w, h ⟦w⟧))) | theorem | Set.ext | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {x y : Set.{u}} : x = y ↔ (∀ z : Set.{u}, z ∈ x ↔ z ∈ y) | ⟨λ h, by simp [h], ext⟩ | theorem | Set.ext_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_injective : function.injective to_set | λ x y h, ext $ set.ext_iff.1 h | theorem | Set.to_set_injective | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_inj {x y : Set} : x.to_set = y.to_set ↔ x = y | to_set_injective.eq_iff | theorem | Set.to_set_inj | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty : Set | mk ∅ | def | Set.empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | The empty ZFC set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_mem_empty (x) : x ∉ (∅ : Set.{u}) | quotient.induction_on x pSet.not_mem_empty | theorem | Set.not_mem_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet.not_mem_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_subset (x : Set.{u}) : (∅ : Set) ⊆ x | quotient.induction_on x $ λ y, subset_iff.2 $ pSet.empty_subset y | theorem | Set.empty_subset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"pSet.empty_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_nonempty_empty : ¬ Set.nonempty ∅ | by simp [Set.nonempty] | theorem | Set.not_nonempty_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_mk_iff {x : pSet} : (mk x).nonempty ↔ x.nonempty | begin
refine ⟨_, λ ⟨a, h⟩, ⟨mk a, h⟩⟩,
rintro ⟨a, h⟩,
induction a using quotient.induction_on,
exact ⟨a, h⟩
end | theorem | Set.nonempty_mk_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_empty (x : Set.{u}) : x = ∅ ↔ ∀ y : Set.{u}, y ∉ x | by { rw ext_iff, simp } | theorem | Set.eq_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_empty_or_nonempty (u : Set) : u = ∅ ∨ u.nonempty | by { rw [eq_empty, ←not_exists], apply em' } | theorem | Set.eq_empty_or_nonempty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"em'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
insert : Set → Set → Set | resp.eval 2 ⟨pSet.insert, λ u v uv ⟨α, A⟩ ⟨β, B⟩ ⟨αβ, βα⟩,
⟨λ o, match o with
| some a := let ⟨b, hb⟩ := αβ a in ⟨some b, hb⟩
| none := ⟨none, uv⟩
end, λ o, match o with
| some b := let ⟨a, ha⟩ := βα b in ⟨some a, ha⟩
| none := ⟨none, uv⟩
end⟩⟩ | def | Set.insert | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | `insert x y` is the set `{x} ∪ y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_insert_iff {x y z : Set.{u}} : x ∈ insert y z ↔ x = y ∨ x ∈ z | quotient.induction_on₃ x y z
(λ x y ⟨α, A⟩, show x ∈ pSet.mk (option α) (λ o, option.rec y A o) ↔
mk x = mk y ∨ x ∈ pSet.mk α A, from
⟨λ m, match m with
| ⟨some a, ha⟩ := or.inr ⟨a, ha⟩
| ⟨none, h⟩ := or.inl (quotient.sound h)
end, λ m, match m with
| or.inr ⟨a, ha⟩ := ⟨some a, ha⟩
| or.inl h := ⟨none,... | theorem | Set.mem_insert_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_insert (x y : Set) : x ∈ insert x y | mem_insert_iff.2 $ or.inl rfl | theorem | Set.mem_insert | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_insert_of_mem {y z : Set} (x) (h : z ∈ y): z ∈ insert x y | mem_insert_iff.2 $ or.inr h | theorem | Set.mem_insert_of_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_insert (x y : Set) : (insert x y).to_set = insert x y.to_set | by { ext, simp } | theorem | Set.to_set_insert | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_singleton {x y : Set.{u}} : x ∈ @singleton Set.{u} Set.{u} _ y ↔ x = y | iff.trans mem_insert_iff ⟨λ o, or.rec (λ h, h) (λ n, absurd n (not_mem_empty _)) o, or.inl⟩ | theorem | Set.mem_singleton | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_singleton (x : Set) : ({x} : Set).to_set = {x} | by { ext, simp } | theorem | Set.to_set_singleton | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
insert_nonempty (u v : Set) : (insert u v).nonempty | ⟨u, mem_insert u v⟩ | theorem | Set.insert_nonempty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_nonempty (u : Set) : Set.nonempty {u} | insert_nonempty u ∅ | theorem | Set.singleton_nonempty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"Set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pair {x y z : Set.{u}} : x ∈ ({y, z} : Set) ↔ x = y ∨ x = z | iff.trans mem_insert_iff $ or_congr iff.rfl mem_singleton | theorem | Set.mem_pair | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega : Set | mk omega | def | Set.omega | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | `omega` is the first infinite von Neumann ordinal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
omega_zero : ∅ ∈ omega | ⟨⟨0⟩, equiv.rfl⟩ | theorem | Set.omega_zero | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega_succ {n} : n ∈ omega.{u} → insert n n ∈ omega.{u} | quotient.induction_on n (λ x ⟨⟨n⟩, h⟩, ⟨⟨n+1⟩, Set.exact $
show insert (mk x) (mk x) = insert (mk $ of_nat n) (mk $ of_nat n), { rw Set.sound h, refl } ⟩) | theorem | Set.omega_succ | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.exact",
"Set.sound"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sep (p : Set → Prop) : Set → Set | resp.eval 1 ⟨pSet.sep (λ y, p (mk y)), λ ⟨α, A⟩ ⟨β, B⟩ ⟨αβ, βα⟩,
⟨λ ⟨a, pa⟩, let ⟨b, hb⟩ := αβ a in ⟨⟨b, by rwa [mk_func, ←Set.sound hb]⟩, hb⟩,
λ ⟨b, pb⟩, let ⟨a, ha⟩ := βα b in ⟨⟨a, by rwa [mk_func, Set.sound ha]⟩, ha⟩⟩⟩ | def | Set.sep | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"Set.sound"
] | `{x ∈ a | p x}` is the set of elements in `a` satisfying `p` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_sep {p : Set.{u} → Prop} {x y : Set.{u}} : y ∈ {y ∈ x | p y} ↔ y ∈ x ∧ p y | quotient.induction_on₂ x y (λ ⟨α, A⟩ y,
⟨λ ⟨⟨a, pa⟩, h⟩, ⟨⟨a, h⟩, by rwa (@quotient.sound pSet _ _ _ h)⟩,
λ ⟨⟨a, h⟩, pa⟩, ⟨⟨a, by { rw mk_func at h, rwa [mk_func, ←Set.sound h] }⟩, h⟩⟩) | theorem | Set.mem_sep | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_sep (a : Set) (p : Set → Prop) :
{x ∈ a | p x}.to_set = {x ∈ a.to_set | p x} | by { ext, simp } | theorem | Set.to_set_sep | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powerset : Set → Set | resp.eval 1 ⟨powerset, λ ⟨α, A⟩ ⟨β, B⟩ ⟨αβ, βα⟩,
⟨λ p, ⟨{b | ∃ a, p a ∧ equiv (A a) (B b)},
λ ⟨a, pa⟩, let ⟨b, ab⟩ := αβ a in ⟨⟨b, a, pa, ab⟩, ab⟩,
λ ⟨b, a, pa, ab⟩, ⟨⟨a, pa⟩, ab⟩⟩,
λ q, ⟨{a | ∃ b, q b ∧ equiv (A a) (B b)},
λ ⟨a, b, qb, ab⟩, ⟨⟨b, qb⟩, ab⟩,
λ ⟨b, qb⟩, let ⟨a, ab⟩ := βα b in ⟨⟨a, b, ... | def | Set.powerset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"equiv"
] | The powerset operation, the collection of subsets of a ZFC set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_powerset {x y : Set.{u}} : y ∈ powerset x ↔ y ⊆ x | quotient.induction_on₂ x y ( λ ⟨α, A⟩ ⟨β, B⟩,
show (⟨β, B⟩ : pSet.{u}) ∈ (pSet.powerset.{u} ⟨α, A⟩) ↔ _,
by simp [mem_powerset, subset_iff]) | theorem | Set.mem_powerset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion_lem {α β : Type u} (A : α → pSet) (B : β → pSet) (αβ : ∀ a, ∃ b, equiv (A a) (B b)) :
∀ a, ∃ b, (equiv ((sUnion ⟨α, A⟩).func a) ((sUnion ⟨β, B⟩).func b)) | | ⟨a, c⟩ := let ⟨b, hb⟩ := αβ a in
begin
induction ea : A a with γ Γ,
induction eb : B b with δ Δ,
rw [ea, eb] at hb,
cases hb with γδ δγ,
exact
let c : type (A a) := c, ⟨d, hd⟩ := γδ (by rwa ea at c) in
have pSet.equiv ((A a).func c) ((B b).func (eq.rec d (eq.symm eb))), from
match A ... | theorem | Set.sUnion_lem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"pSet",
"pSet.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion : Set → Set | resp.eval 1 ⟨pSet.sUnion, λ ⟨α, A⟩ ⟨β, B⟩ ⟨αβ, βα⟩,
⟨sUnion_lem A B αβ, λ a, exists.elim (sUnion_lem B A (λ b,
exists.elim (βα b) (λ c hc, ⟨c, pSet.equiv.symm hc⟩)) a) (λ b hb, ⟨b, pSet.equiv.symm hb⟩)⟩⟩ | def | Set.sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"pSet.equiv.symm"
] | The union operator, the collection of elements of elements of a ZFC set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sInter (x : Set) : Set | by { classical, exact dite x.nonempty (λ h, {y ∈ h.some | ∀ z ∈ x, y ∈ z}) (λ _, ∅) } | def | Set.sInter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | The intersection operator, the collection of elements in all of the elements of a ZFC set. We
special-case `⋂₀ ∅ = ∅`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_sUnion {x y : Set.{u}} : y ∈ ⋃₀ x ↔ ∃ z ∈ x, y ∈ z | quotient.induction_on₂ x y (λ x y, iff.trans mem_sUnion
⟨λ ⟨z, h⟩, ⟨⟦z⟧, h⟩, λ ⟨z, h⟩, quotient.induction_on z (λ z h, ⟨z, h⟩) h⟩) | theorem | Set.mem_sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sInter {x y : Set} (h : x.nonempty) : y ∈ ⋂₀ x ↔ ∀ z ∈ x, y ∈ z | begin
rw [sInter, dif_pos h],
simp only [mem_to_set, mem_sep, and_iff_right_iff_imp],
exact λ H, H _ h.some_mem
end | theorem | Set.mem_sInter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"and_iff_right_iff_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion_empty : ⋃₀ (∅ : Set) = ∅ | by { ext, simp } | theorem | Set.sUnion_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sInter_empty : ⋂₀ (∅ : Set) = ∅ | dif_neg $ by simp | theorem | Set.sInter_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_mem_sInter {x y z : Set} (hy : y ∈ ⋂₀ x) (hz : z ∈ x) : y ∈ z | begin
rcases eq_empty_or_nonempty x with rfl | hx,
{ exact (not_mem_empty z hz).elim },
{ exact (mem_sInter hx).1 hy z hz }
end | theorem | Set.mem_of_mem_sInter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sUnion_of_mem {x y z : Set} (hy : y ∈ z) (hz : z ∈ x) : y ∈ ⋃₀ x | mem_sUnion.2 ⟨z, hz, hy⟩ | theorem | Set.mem_sUnion_of_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_sInter_of_not_mem {x y z : Set} (hy : ¬ y ∈ z) (hz : z ∈ x) : ¬ y ∈ ⋂₀ x | λ hx, hy $ mem_of_mem_sInter hx hz | theorem | Set.not_mem_sInter_of_not_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion_singleton {x : Set.{u}} : ⋃₀ ({x} : Set) = x | ext $ λ y, by simp_rw [mem_sUnion, exists_prop, mem_singleton, exists_eq_left] | theorem | Set.sUnion_singleton | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"exists_eq_left",
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sInter_singleton {x : Set.{u}} : ⋂₀ ({x} : Set) = x | ext $ λ y, by simp_rw [mem_sInter (singleton_nonempty x), mem_singleton, forall_eq] | theorem | Set.sInter_singleton | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"forall_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_sUnion (x : Set.{u}) : (⋃₀ x).to_set = ⋃₀ (to_set '' x.to_set) | by { ext, simp } | theorem | Set.to_set_sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_sInter {x : Set.{u}} (h : x.nonempty) : (⋂₀ x).to_set = ⋂₀ (to_set '' x.to_set) | by { ext, simp [mem_sInter h] } | theorem | Set.to_set_sInter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_injective : function.injective (@singleton Set Set _) | λ x y H, let this := congr_arg sUnion H in by rwa [sUnion_singleton, sUnion_singleton] at this | theorem | Set.singleton_injective | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_inj {x y : Set} : ({x} : Set) = {y} ↔ x = y | singleton_injective.eq_iff | theorem | Set.singleton_inj | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union (x y : Set.{u}) : Set.{u} | ⋃₀ {x, y} | def | Set.union | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | The binary union operation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inter (x y : Set.{u}) : Set.{u} | {z ∈ x | z ∈ y} | def | Set.inter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | The binary intersection operation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diff (x y : Set.{u}) : Set.{u} | {z ∈ x | z ∉ y} | def | Set.diff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | The set difference operation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_set_union (x y : Set.{u}) : (x ∪ y).to_set = x.to_set ∪ y.to_set | by { unfold has_union.union, rw Set.union, simp } | theorem | Set.to_set_union | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_inter (x y : Set.{u}) : (x ∩ y).to_set = x.to_set ∩ y.to_set | by { unfold has_inter.inter, rw Set.inter, ext, simp } | theorem | Set.to_set_inter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_sdiff (x y : Set.{u}) : (x \ y).to_set = x.to_set \ y.to_set | by { change {z ∈ x | z ∉ y}.to_set = _, ext, simp } | theorem | Set.to_set_sdiff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_union {x y z : Set.{u}} : z ∈ x ∪ y ↔ z ∈ x ∨ z ∈ y | by { rw ←mem_to_set, simp } | theorem | Set.mem_union | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inter {x y z : Set.{u}} : z ∈ x ∩ y ↔ z ∈ x ∧ z ∈ y | @@mem_sep (λ z : Set.{u}, z ∈ y) | theorem | Set.mem_inter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_diff {x y z : Set.{u}} : z ∈ x \ y ↔ z ∈ x ∧ z ∉ y | @@mem_sep (λ z : Set.{u}, z ∉ y) | theorem | Set.mem_diff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion_pair {x y : Set.{u}} : ⋃₀ ({x, y} : Set.{u}) = x ∪ y | begin
ext,
simp_rw [mem_union, mem_sUnion, mem_pair],
split,
{ rintro ⟨w, (rfl | rfl), hw⟩,
{ exact or.inl hw },
{ exact or.inr hw } },
{ rintro (hz | hz),
{ exact ⟨x, or.inl rfl, hz⟩ },
{ exact ⟨y, or.inr rfl, hz⟩ } }
end | theorem | Set.sUnion_pair | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_wf : @well_founded Set (∈) | well_founded_lift₂_iff.mpr pSet.mem_wf | theorem | Set.mem_wf | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"pSet.mem_wf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on {p : Set → Prop} (x) (h : ∀ x, (∀ y ∈ x, p y) → p x) : p x | mem_wf.induction x h | theorem | Set.induction_on | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | Induction on the `∈` relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_asymm {x y : Set} : x ∈ y → y ∉ x | asymm | theorem | Set.mem_asymm | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_irrefl (x : Set) : x ∉ x | irrefl x | theorem | Set.mem_irrefl | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regularity (x : Set.{u}) (h : x ≠ ∅) : ∃ y ∈ x, x ∩ y = ∅ | classical.by_contradiction $ λ ne, h $ (eq_empty x).2 $ λ y,
induction_on y $ λ z (IH : ∀ w : Set.{u}, w ∈ z → w ∉ x), show z ∉ x, from λ zx,
ne ⟨z, zx, (eq_empty _).2 (λ w wxz, let ⟨wx, wz⟩ := mem_inter.1 wxz in IH w wz wx)⟩ | theorem | Set.regularity | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image (f : Set → Set) [H : definable 1 f] : Set → Set | let r := @definable.resp 1 f _ in
resp.eval 1 ⟨image r.1, λ x y e, mem.ext $ λ z,
iff.trans (mem_image r.2) $ iff.trans (by exact
⟨λ ⟨w, h1, h2⟩, ⟨w, (mem.congr_right e).1 h1, h2⟩,
λ ⟨w, h1, h2⟩, ⟨w, (mem.congr_right e).2 h1, h2⟩⟩) $
iff.symm (mem_image r.2)⟩ | def | Set.image | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | The image of a (definable) ZFC set function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.mk :
Π (f : Set.{u} → Set.{u}) [H : definable 1 f] (x) {y} (h : y ∈ x), f y ∈ @image f H x | | ._ ⟨F⟩ x y := quotient.induction_on₂ x y $ λ ⟨α, A⟩ y ⟨a, ya⟩, ⟨a, F.2 _ _ ya⟩ | theorem | Set.image.mk | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_image : Π {f : Set.{u} → Set.{u}} [H : definable 1 f] {x y : Set.{u}},
y ∈ @image f H x ↔ ∃ z ∈ x, f z = y | | ._ ⟨F⟩ x y := quotient.induction_on₂ x y $ λ ⟨α, A⟩ y,
⟨λ ⟨a, ya⟩, ⟨⟦A a⟧, mem.mk A a, eq.symm $ quotient.sound ya⟩,
λ ⟨z, hz, e⟩, e ▸ image.mk _ _ hz⟩ | theorem | Set.mem_image | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_image (f : Set → Set) [H : definable 1 f] (x : Set) :
(image f x).to_set = f '' x.to_set | by { ext, simp } | theorem | Set.to_set_image | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range {α : Type u} (f : α → Set.{max u v}) : Set.{max u v} | ⟦⟨ulift α, quotient.out ∘ f ∘ ulift.down⟩⟧ | def | Set.range | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"quotient.out"
] | The range of an indexed family of sets. The universes allow for a more general index type
without manual use of `ulift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_range {α : Type u} {f : α → Set.{max u v}} {x : Set.{max u v}} :
x ∈ range f ↔ x ∈ set.range f | quotient.induction_on x (λ y, begin
split,
{ rintro ⟨z, hz⟩,
exact ⟨z.down, quotient.eq_mk_iff_out.2 hz.symm⟩ },
{ rintro ⟨z, hz⟩,
use z,
simpa [hz] using pSet.equiv.symm (quotient.mk_out y) }
end) | theorem | Set.mem_range | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet.equiv.symm",
"quotient.mk_out",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_range {α : Type u} (f : α → Set.{max u v}) :
(range f).to_set = set.range f | by { ext, simp } | theorem | Set.to_set_range | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair (x y : Set.{u}) : Set.{u} | {{x}, {x, y}} | def | Set.pair | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Kuratowski ordered pair | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_set_pair (x y : Set.{u}) : (pair x y).to_set = {{x}, {x, y}} | by simp [pair] | theorem | Set.to_set_pair | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair_sep (p : Set.{u} → Set.{u} → Prop) (x y : Set.{u}) : Set.{u} | {z ∈ powerset (powerset (x ∪ y)) | ∃ a ∈ x, ∃ b ∈ y, z = pair a b ∧ p a b} | def | Set.pair_sep | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | A subset of pairs `{(a, b) ∈ x × y | p a b}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_pair_sep {p} {x y z : Set.{u}} :
z ∈ pair_sep p x y ↔ ∃ a ∈ x, ∃ b ∈ y, z = pair a b ∧ p a b | begin
refine mem_sep.trans ⟨and.right, λ e, ⟨_, e⟩⟩,
rcases e with ⟨a, ax, b, bY, rfl, pab⟩,
simp only [mem_powerset, subset_def, mem_union, pair, mem_pair],
rintros u (rfl|rfl) v; simp only [mem_singleton, mem_pair],
{ rintro rfl, exact or.inl ax },
{ rintro (rfl|rfl); [left, right]; assumption }
end | theorem | Set.mem_pair_sep | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair_injective : function.injective2 pair | λ x x' y y' H, begin
have ae := ext_iff.1 H,
simp only [pair, mem_pair] at ae,
obtain rfl : x = x',
{ cases (ae {x}).1 (by simp) with h h,
{ exact singleton_injective h },
{ have m : x' ∈ ({x} : Set),
{ simp [h] },
rw mem_singleton.mp m } },
have he : x = y → y = y',
{ rintro rfl,
ca... | theorem | Set.pair_injective | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair_inj {x y x' y' : Set} : pair x y = pair x' y' ↔ x = x' ∧ y = y' | pair_injective.eq_iff | theorem | Set.pair_inj | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod : Set.{u} → Set.{u} → Set.{u} | pair_sep (λ a b, true) | def | Set.prod | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | The cartesian product, `{(a, b) | a ∈ x, b ∈ y}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_prod {x y z : Set.{u}} : z ∈ prod x y ↔ ∃ a ∈ x, ∃ b ∈ y, z = pair a b | by simp [prod] | theorem | Set.mem_prod | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair_mem_prod {x y a b : Set.{u}} : pair a b ∈ prod x y ↔ a ∈ x ∧ b ∈ y | ⟨λ h, let ⟨a', a'x, b', b'y, e⟩ := mem_prod.1 h in
match a', b', pair_injective e, a'x, b'y with ._, ._, ⟨rfl, rfl⟩, ax, bY := ⟨ax, bY⟩ end,
λ ⟨ax, bY⟩, mem_prod.2 ⟨a, ax, b, bY, rfl⟩⟩ | theorem | Set.pair_mem_prod | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_func (x y f : Set.{u}) : Prop | f ⊆ prod x y ∧ ∀ z : Set.{u}, z ∈ x → ∃! w, pair z w ∈ f | def | Set.is_func | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | `is_func x y f` is the assertion that `f` is a subset of `x × y` which relates to each element
of `x` a unique element of `y`, so that we can consider `f`as a ZFC function `x → y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
funs (x y : Set.{u}) : Set.{u} | {f ∈ powerset (prod x y) | is_func x y f} | def | Set.funs | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | `funs x y` is `y ^ x`, the set of all set functions `x → y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_funs {x y f : Set.{u}} : f ∈ funs x y ↔ is_func x y f | by simp [funs, is_func] | theorem | Set.mem_funs | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_definable_aux (f : Set → Set) [H : definable 1 f] :
definable 1 (λ y, pair y (f y)) | @classical.all_definable 1 _ | instance | Set.map_definable_aux | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"classical.all_definable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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