(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
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(* File: axs_fundation.v                                                   *)
(* auteur: alex@sysal.ibp.fr                                               *)
(* date: 19/10/95                                                          *)
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Require Export axs_choice.
Axiom
axs_fundation :
forall x : E, x <> Vide -> exists y : E, In y x /\ inter y x = Vide.
(* Proof of lem_sing_not_empty                                             *)
unfold not in |- *; intros.
unfold Vide in H; generalize (lem_vide_propertie x var_vide); rewrite <- H;
intros.
generalize (lem_x_in_sing_x x); intros.
absurd (In x (singleton x)); auto with zfc.
Qed.
(* End of proof of lem_sing_not_empty                                      *)

Lemma lem_x_not_in_x : forall x : E, ~ In x x.