(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 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.

(* Specifications: *)
Theorem IN_Class_succ : forall E : Ens, IN E (Class_succ E).
intros E; unfold Class_succ in |- *; unfold Sing in |- *;
apply IN_Union with (Paire E E); auto with zfc.