(* The set of small sets                        *)

Definition Big := sup Ens' inj.

(* Specifications: *)
Theorem Big_is_big : forall E : Ens', IN (inj E) Big.
Theorem IN_Big_small : forall E : Ens, IN E Big -> EXType' _ (fun E' : Ens' => EQ E (inj E')).
Theorem IN_small_small : forall (E : Ens) (E' : Ens'), IN E (inj E') -> EXType' _ (fun E1 : Ens' => EQ E (inj E1)).
Theorem Big_closed_for_power : forall E : Ens, IN E Big -> IN (Power E) Big.
Theorem Russell : forall E : Ens, (forall E' : Ens, IN E' E) -> False.
Theorem Big_is_big : forall E : Ens', IN (inj E) Big.
Theorem IN_Big_small : forall E : Ens, IN E Big -> EXType' _ (fun E' : Ens' => EQ E (inj E')).
Theorem IN_small_small : forall (E : Ens) (E' : Ens'), IN E (inj E') -> EXType' _ (fun E1 : Ens' => EQ E (inj E1)).
Theorem Big_closed_for_power : forall E : Ens, IN E Big -> IN (Power E) Big.