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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* The set of natural numbers  *)
Definition Class_succ (E : Ens) := Union (Paire E (Sing E)).
(* Inductive Ord : Ens -> Prop := Oo : (Ord Vide) | So : (E:Ens)(Ord E)->(Ord (Class_succ E)) | Lo : (E:Ens)((e:Ens)(IN e E)->(Ord e))->(Ord (Union E)) | Eo : (E,E':Ens)(Ord E)->(EQ E E')->(Ord E').  Hints Resolve Oo So Lo : zfc. *)
Definition Nat : nat -> Ens.
simple induction 1; intros.
exact Vide.
exact (Class_succ X).
Defined.
(* Theorem Nat_Ord : (n:nat)(Ord (Nat n)). Induction n; Simpl; Auto with zfc. Save. *)
Definition Omega : Ens := sup nat Nat.
intros E; unfold Class_succ in |- *; unfold Sing in |- *;
apply IN_Union with (Paire E E); auto with zfc.
Qed.
unfold INC in |- *; unfold Class_succ in |- *.
intros.
apply IN_Union with E; auto with zfc.
Qed.
Hint Resolve IN_Class_succ INC_Class_succ: zfc.

Theorem IN_Class_succ_or :
forall E E' : Ens, IN E' (Class_succ E) -> EQ E E' \/ IN E' E.