(* Proof of lem_antisym_not_sym                                            *)
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 (H x x0 H2 (H0 x x0 H2)); intros.
absurd (x = x0); auto with zfc.
Qed.
(* End of proof of lem_antisym_not_sym                                     *)

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(* Une relation irreflexive et transitive est antisymmetrique              *)
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Lemma lem_irrefl_trans_so_antisym :
forall (r a : E) (rRel : rel_sig r a a) (rIrrefl : irreflexivity r a rRel)
(rTrans : transitivity r a rRel), antisymmetry r a rRel.