(* End of proof of lem_not_vide_in_vide                                    *)

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(* Intersection de deux ensembles                                          *)
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Definition def_F_inter (s x : E) := In x s.Definition inter (a b : E) := subset (def_F_inter b) a.

(* Specifications: *)
Lemma lem_inter_propertie : forall a b x : E, In x (inter a b) <-> In x a /\ In x b.
Lemma lem_inter_propertie_lr : forall a b x : E, In x (inter a b) -> In x a /\ In x b.
Lemma lem_inter_propertie_rl : forall a b x : E, In x a /\ In x b -> In x (inter a b).
Lemma lem_inter_unicite : forall a b i : E, (forall x : E, In x i <-> In x a /\ In x b) -> i = inter a b.
Lemma lem_inter_vide_prop : forall x : E, inter x Vide = Vide.
Theorem notEEexists : ~ (exists v0 : E, (forall v1 : E, In v1 v0)).