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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* Definitions of the empty set, pair, union, intersection, comprehension  *)
(*  axiom and powerset, together with their properties                     *)
(* Useful types (actually top and bottom)   *)
Inductive Un : Set :=
void : Un.
Inductive F : Set :=.
(* The empty set  (vide = french for empty)   *)
Definition Vide : Ens := sup F (fun f : F => match f return Ens with
end).
(* The axioms of the empty set *)
unfold Vide in |- *; simpl in |- *; intros E H; cut False.
simple induction 1.
elim H; intros x; elim x.
Qed.

Theorem tout_vide_est_Vide :
forall E : Ens, (forall E' : Ens, IN E' E -> F) -> EQ E Vide.