Axiom axs_vide_or_not : forall x : E, x = Vide \/ x <> Vide.Axiom
  axs_fundation :
    forall x : E, x <> Vide -> exists y : E, In y x /\ inter y x = Vide.

(* Specifications: *)
Lemma lem_sing_not_empty : forall x : E, singleton x <> Vide.
Lemma lem_x_not_in_x : forall x : E, ~ In x x.
Lemma lem_inter_is_empty : forall a b : E, (forall x : E, In x a -> ~ In x b) -> (forall y : E, In y b -> ~ In y a) -> inter a b = Vide.