(* Proof of lem_sym_not_antisym                                            *)
unfold not in |- *; unfold symmetry in |- *; unfold antisymmetry in |- *;
intros.
elim rNe; clear rNe; intros; elim H1; clear H1; intros; elim H1; clear H1;
intros.
generalize (H0 x x0 H2 (H x x0 H2)); intros.
absurd (x = x0); auto with zfc.
Qed.
(* End of proof of lem_sym_not_antisym                                     *)

(***************************************************************************)
(* Une relation antisymmetrique dont le domaine contient au moins deux     *)
(* elements distincts n'est pas symmetrique.                               *)
(***************************************************************************)
Lemma lem_antisym_not_sym :
forall (r a : E) (rRel : rel_sig r a a)
(rNe : exists x : E, (exists y : E, x <> y /\ In (couple x y) r)),
antisymmetry r a rRel -> ~ symmetry r a rRel.