(* Proof of lem_in_singleton                                               *)
unfold singleton in |- *; unfold iff in |- *; split; intros;
[ elim H; intros; auto with zfc | apply H0; left; auto with zfc ].
Qed.
(* End of proof of lem_in_singleton                                        *)

Lemma lem_x_in_sing_x : forall x : E, In x (singleton x).
intros; unfold singleton in |- *; elim (axs_paire x x x); intros; apply H0;
clear H0 H; left; reflexivity.