intros E X o Y o1 inc inc1.
apply IN_Vee with Y.
auto with zfc.
unfold Succ in |- *; apply IN_P_Comp; auto with zfc.
intros; apply Ord_sound with w1; auto with zfc.
auto with zfc.
apply INC_IN_Power; auto with zfc.
auto with zfc.
Qed.
Inductive depprod1 (A : Type) (P : A -> Type) : Type :=
dep_i1 : forall x : A, P x -> depprod1 A P.
Definition PI11 (A : Type) (P : A -> Type) (c : depprod1 A P) :=
match c with
| dep_i1 a p => a
end.
Definition PI21 (A : Type) (P : A -> Type) (c : depprod1 A P) :=
match c return (P (PI11 A P c)) with
| dep_i1 a p => p
end.

(* Specifications: *)
Lemma ex_rk :
forall E : Ens, depprod1 _ (fun L : Ens => Ord L /\ IN E (Vee L)).