(* Proof of lem_singleton_propertie                                        *)
intros; unfold singleton in |- *.
elim (lem_paire_propertie a a b b); intros.
unfold iff in |- *; split; intros.
apply lem_and_contract; apply lem_or_contract; auto with zfc.
rewrite H1; reflexivity.
Qed.
(* End of proof of lem_singleton_propertie                                 *)

Lemma lem_in_singleton : forall a x : E, In x (singleton a) <-> x = a.
intros; elim (axs_paire a a x); intros.