import Mathlib

namespace Chapter3

-- <CONTEXT>
universe u v

class SetTheory where
  Set : Type u
  Object : Type v
  set_to_object : Set ↪ Object
  mem : Object → Set → Prop
  extensionality X Y : (∀ x, mem x X ↔ mem x Y) → X = Y
  emptyset: Set
  emptyset_mem x : ¬ mem x emptyset
  singleton : Object → Set
  singleton_axiom x y : mem x (singleton y) ↔ x = y
  union_pair : Set → Set → Set
  union_pair_axiom X Y x : mem x (union_pair X Y) ↔ (mem x X ∨ mem x Y)
  specify A (P: Subtype (mem . A) → Prop) : Set
  specification_axiom A (P: Subtype (mem . A) → Prop) :
    (∀ x, mem x (specify A P) → mem x A) ∧ ∀ x, mem x.val (specify A P) ↔ P x
  replace A (P: Subtype (mem . A) → Object → Prop)
    (hP: ∀ x y y', P x y ∧ P x y' → y = y') : Set
  replacement_axiom A (P: Subtype (mem . A) → Object → Prop)
    (hP: ∀ x y y', P x y ∧ P x y' → y = y') : ∀ y, mem y (replace A P hP) ↔ ∃ x, P x y
  nat : Set
  nat_equiv : ℕ ≃ Subtype (mem . nat)
  regularity_axiom A (hA : ∃ x, mem x A) :
    ∃ x, mem x A ∧ ∀ S, x = set_to_object S → ¬ ∃ y, mem y A ∧ mem y S
  pow : Set → Set → Set
  function_to_object (X: Set) (Y: Set) :
    (Subtype (mem . X) → Subtype (mem . Y)) ↪ Object
  powerset_axiom (X: Set) (Y: Set) (F:Object) :
    mem F (pow X Y) ↔ ∃ f: Subtype (mem . Y) → Subtype (mem . X),
    function_to_object Y X f = F
  union : Set → Set
  union_axiom A x : mem x (union A) ↔ ∃ S, mem x S ∧ mem (set_to_object S) A

export SetTheory (Set Object)

variable [SetTheory]

instance SetTheory.objects_mem_sets : Membership Object Set where
  mem X x := mem x X

instance SetTheory.sets_are_objects : Coe Set Object where
  coe X := set_to_object X

theorem SetTheory.Set.coe_eq {X Y:Set} (h: (X: Object) = (Y: Object)) : X = Y := by
  sorry

@[simp]
theorem SetTheory.Set.coe_eq_iff (X Y:Set) : (X: Object) = (Y: Object) ↔  X = Y := by
  sorry

@[ext]
theorem SetTheory.Set.ext {X Y:Set} (h: ∀ x, x ∈ X ↔ x ∈ Y) : X = Y := by
  sorry

instance SetTheory.Set.instEmpty : EmptyCollection Set where
  emptyCollection := emptyset

open SetTheory.Set

@[simp]
theorem SetTheory.Set.not_mem_empty : ∀ x, x ∉ (∅:Set) := by
  sorry

theorem SetTheory.Set.eq_empty_iff_forall_notMem {X:Set} : X = ∅ ↔ (∀ x, x ∉ X) := by
  sorry

theorem SetTheory.Set.empty_unique : ∃! (X:Set), ∀ x, x ∉ X := by
  sorry

lemma SetTheory.Set.nonempty_def {X:Set} (h: X ≠ ∅) : ∃ x, x ∈ X := by
  sorry

theorem SetTheory.Set.nonempty_of_inhabited {X:Set} {x:Object} (h:x ∈ X) : X ≠ ∅ := by
  sorry

instance SetTheory.Set.instSingleton : Singleton Object Set where
  singleton := singleton

@[simp]
theorem SetTheory.Set.mem_singleton (x a:Object) : x ∈ ({a}:Set) ↔ x = a := by
  sorry

instance SetTheory.Set.instUnion : Union Set where
  union := union_pair

@[simp]
theorem SetTheory.Set.mem_union (x:Object) (X Y:Set) : x ∈ (X ∪ Y) ↔ (x ∈ X ∨ x ∈ Y) := by
  sorry

instance SetTheory.Set.instInsert : Insert Object Set where
  insert x X := {x} ∪ X

@[simp]
theorem SetTheory.Set.mem_insert (a b: Object) (X: Set) : a ∈ insert b X ↔ a = b ∨ a ∈ X := by
  sorry

theorem SetTheory.Set.pair_eq (a b:Object) : ({a,b}:Set) = {a} ∪ {b} := by
  sorry

@[simp]
theorem SetTheory.Set.mem_pair (x a b:Object) : x ∈ ({a,b}:Set) ↔ (x = a ∨ x = b) := by
  sorry

@[simp]
theorem SetTheory.Set.mem_triple (x a b c:Object) : x ∈ ({a,b,c}:Set) ↔ (x = a ∨ x = b ∨ x = c) := by
  sorry

theorem SetTheory.Set.singleton_uniq (a:Object) : ∃! (X:Set), ∀ x, x ∈ X ↔ x = a := by
  sorry

theorem SetTheory.Set.pair_uniq (a b:Object) : ∃! (X:Set), ∀ x, x ∈ X ↔ x = a ∨ x = b := by
  sorry

theorem SetTheory.Set.pair_comm (a b:Object) : ({a,b}:Set) = {b,a} := by
  sorry

@[simp]
theorem SetTheory.Set.pair_self (a:Object) : ({a,a}:Set) = {a} := by
  sorry

abbrev SetTheory.Set.empty : Set := ∅
abbrev SetTheory.Set.singleton_empty : Set := {(empty: Object)}
abbrev SetTheory.Set.pair_empty : Set := {(empty: Object), (singleton_empty: Object)}

theorem SetTheory.Set.union_congr_left (A A' B:Set) (h: A = A') : A ∪ B = A' ∪ B := by
  sorry

theorem SetTheory.Set.union_congr_right (A B B':Set) (h: B = B') : A ∪ B = A ∪ B' := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem SetTheory.Set.singleton_union_singleton (a b:Object) :
    ({a}:Set) ∪ ({b}:Set) = {a,b} := by
-- </TARGET_THEOREM>

end Chapter3