| import Mathlib | |
| open CategoryTheory Functor Limits | |
| namespace CAT_statement_S_0032 | |
| def IsIsoClosed (P : Type u → Prop) : Prop := | |
| ∀ {X Y : Type u}, Nonempty (X ≅ Y) → P X → P Y | |
| def SubcategoryEquiv (P Q : Type u → Prop) : Prop := | |
| ∀ X, P X ↔ Q X | |
| def IsReflectiveSubcategory (P : Type u → Prop) : Prop := | |
| Nonempty (Reflective (ObjectProperty.ι P)) | |
| theorem Set_has_precisely_three_reflective_subcategories : | |
| ∃ (P₁ P₂ P₃ : Type u → Prop), | |
| IsIsoClosed P₁ ∧ IsReflectiveSubcategory P₁ ∧ | |
| IsIsoClosed P₂ ∧ IsReflectiveSubcategory P₂ ∧ | |
| IsIsoClosed P₃ ∧ IsReflectiveSubcategory P₃ ∧ | |
| ¬ SubcategoryEquiv P₁ P₂ ∧ ¬ SubcategoryEquiv P₂ P₃ ∧ ¬ SubcategoryEquiv P₁ P₃ ∧ | |
| ∀ (Q : Type u → Prop), IsIsoClosed Q → IsReflectiveSubcategory Q → | |
| (SubcategoryEquiv Q P₁ ∨ SubcategoryEquiv Q P₂ ∨ SubcategoryEquiv Q P₃) := by | |
| sorry | |
| end CAT_statement_S_0032 | |