Section Ensembles.
Variable U : Type.
Definition Ensemble := U -> Prop.
Definition In (A:Ensemble) (x:U) : Prop := A x.
Definition Included (B C:Ensemble) : Prop := forall x:U, In B x -> In C x.
Inductive Empty_set : Ensemble :=.
Inductive Full_set : Ensemble :=
Full_intro : forall x:U, In Full_set x.
(** NB: The following definition builds-in equality of elements in [U] as Leibniz equality.  This may have to be changed if we replace [U] by a Setoid on [U] with its own equality [eqs], with [In_singleton: (y: U)(eqs x y) -> (In (Singleton x) y)]. *)
Inductive Singleton (x:U) : Ensemble :=
In_singleton : In (Singleton x) x.
Inductive Union (B C:Ensemble) : Ensemble :=
| Union_introl : forall x:U, In B x -> In (Union B C) x
| Union_intror : forall x:U, In C x -> In (Union B C) x.
Definition Add (B:Ensemble) (x:U) : Ensemble := Union B (Singleton x).
Inductive Intersection (B C:Ensemble) : Ensemble :=
Intersection_intro :
forall x:U, In B x -> In C x -> In (Intersection B C) x.
Inductive Couple (x y:U) : Ensemble :=
| Couple_l : In (Couple x y) x
| Couple_r : In (Couple x y) y.
Inductive Triple (x y z:U) : Ensemble :=
| Triple_l : In (Triple x y z) x
| Triple_m : In (Triple x y z) y
| Triple_r : In (Triple x y z) z.
Definition Complement (A:Ensemble) : Ensemble := fun x:U => ~ In A x.
Definition Setminus (B C:Ensemble) : Ensemble :=
fun x:U => In B x /\ ~ In C x.
Definition Subtract (B:Ensemble) (x:U) : Ensemble := Setminus B (Singleton x).
Inductive Disjoint (B C:Ensemble) : Prop :=
Disjoint_intro : (forall x:U, ~ In (Intersection B C) x) -> Disjoint B C.
Inductive Inhabited (B:Ensemble) : Prop :=
Inhabited_intro : forall x:U, In B x -> Inhabited B.
Definition Strict_Included (B C:Ensemble) : Prop := Included B C /\ B <> C.
Definition Same_set (B C:Ensemble) : Prop := Included B C /\ Included C B.
(** Extensionality Axiom *)

Axiom Extensionality_Ensembles : forall A B:Ensemble, Same_set A B -> A = B.