Definition Omega : Ens := sup nat Nat.

(* Specifications: *)
Theorem IN_Succ : forall E : Ens, IN E (Succ E).
Theorem INC_Succ : forall E : Ens, INC E (Succ E).
Theorem IN_Succ_or : forall E E' : Ens, IN E' (Succ E) -> EQ E E' \/ IN E' E.
Theorem E_not_IN_E : forall E : Ens, IN E E -> F.
Theorem Nat_IN_Omega : forall n : nat, IN (Nat n) Omega.
Theorem IN_Omega_EXType : forall E : Ens, IN E Omega -> EXType _ (fun n : nat => EQ (Nat n) E).
Theorem IN_Nat_EXType : forall (n : nat) (E : Ens), IN E (Nat n) -> EXType _ (fun p : nat => EQ E (Nat p)).
Theorem Omega_EQ_Union : EQ Omega (Union Omega).
Theorem Omega_Ord : Ord Omega.