unfold Succ in i.
elim (Union_IN (Paire E (Sing E)) E' i).
intros E1; simple induction 1; intros i1 i2.
elim (Paire_IN E (Sing E) E1 i1).
intros; right; apply IN_sound_right with E1; auto with zfc.
intros; left; cut (IN E' (Sing E)).
auto with zfc.
apply IN_sound_right with E1; auto with zfc.
Qed.

(* Specifications: *)
Theorem E_not_IN_E : forall E : Ens, IN E E -> F.
simple induction E; intros A f r i.