(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(***************************************************************************)
(* File: axs_comprehension.v                                               *)
(* auteur: alex@sysal.ibp.fr                                               *)
(* date: 19/10/95                                                          *)
(***************************************************************************)
Require Export axs_parties.
(***************************************************************************)
(* Schema d'axiome de comprehension                                        *)
(***************************************************************************)
Parameter subset : (E -> Prop) -> E -> E.
Axiom
axs_comprehension :
forall (F : E -> Prop) (a v0 : E), In v0 (subset F a) <-> In v0 a /\ F v0.
(* Proof of lem_comprehension_lr                                           *)
intros; elim (axs_comprehension F a v0); intros; exact (H0 H).
Qed.
(* End of proof of lem_comprehension_lr                                    *)

Lemma lem_comprehension_rl :
forall (F : E -> Prop) (a v0 : E), In v0 a /\ F v0 -> In v0 (subset F a).