intros.
elim (IN_Omega_EXType E H); intros n e.
apply IN_Union with (Nat (S n)).
auto with zfc.
apply IN_sound_left with (Nat n).
auto with zfc.
auto with zfc.
change (IN (Nat n) (Succ (Nat n))) in |- *; auto with zfc.
intros.
elim (Union_IN Omega E H).
intros e h.
elim h.
intros i1 i2.
elim (IN_Omega_EXType e i1).
intros n e1.
cut (IN E (Nat n)).
intros.
elim (IN_Nat_EXType n E H0); intros.
apply IN_sound_left with (Nat x); auto with zfc.
apply IN_sound_right with e; auto with zfc.
Qed.

(* Specifications: *)
Theorem Omega_Ord : Ord Omega.
apply Eo with (Union Omega).