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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* This is a step towards inaccessible cardinals                *)
(* We can define "small" sets, using the Type's hierarchy       *)
(* Since Coq does not support universe polymorphism, we need    *)
(* to duplicate some definitions here.                          *)
(* The small sets  *)
Inductive Ens' : Type :=
sup' : forall A : Type, (A -> Ens') -> Ens'.
(* Existential quantification *)
Inductive EXType' (P : Type) (Q : P -> Prop) : Prop :=
EXTypei' : forall x : P, Q x -> EXType' P Q.
(* Cartesian product in Type *)
Inductive prod_t' (A B : Type) : Type :=
pair_t' : A -> B -> prod_t' A B.
(* Existential on the Type level *)
Inductive depprod'' (A : Type) (P : A -> Type) : Type :=
dep_i'' : forall x : A, P x -> depprod'' A P.
(* Recursive Definition of the extentional equality on sets *)
Definition EQ' : Ens' -> Ens' -> Prop.
simple induction 1; intros A f eq1.
simple induction 1; intros B g eq2.
apply and.
exact (forall x : A, EXType' _ (fun y : B => eq1 x (g y))).
exact (forall y : B, EXType' _ (fun x : A => eq1 x (g y))).
Defined.
(* small sets can be injected into big sets             *)
Definition inj : Ens' -> Ens.
simple induction 1; intros A f fr.
exact (sup A fr).
Defined.
simple induction E1; intros A1 f1 fr1; simple induction E2; intros A2 f2 r2;
simpl in |- *.
simple induction 1; intros HR1 HR2; split.
intros a1; elim (HR1 a1); intros a2 Ha2; exists a2; auto with zfc.
intros a2; elim (HR2 a2); intros a1 Ha1; exists a1; auto with zfc.
Qed.
Definition Power' (E : Ens') : Ens' :=
match E with
| sup' A f =>
sup' _
(fun P : A -> Prop =>
sup' _
(fun c : depprod'' A (fun a : A => P a) =>
match c with
| dep_i'' a p => f a
end))
end.
simple induction E; intros A f HR.
simpl in |- *; split.
intros P; exists P; split.
intros c; elim c; intros a p.
exists (dep_i A (fun a0 : A => P a0) a p); simpl in |- *; auto with zfc.
intros c; elim c; intros a p.
exists (dep_i'' A (fun a0 : A => P a0) a p); simpl in |- *; auto with zfc.
intros P; exists P; split.
intros c; elim c; intros a p.
exists (dep_i A (fun a0 : A => P a0) a p); simpl in |- *; auto with zfc.
intros c; elim c; intros a p.
exists (dep_i'' A (fun a0 : A => P a0) a p); simpl in |- *; auto with zfc.
Qed.
(* The set of small sets                        *)
Definition Big := sup Ens' inj.
intros E; unfold Big in |- *; simpl in |- *; exists E; auto with zfc.
Qed.
unfold Big in |- *; simpl in |- *; simple induction 1; intros E' HE';
exists E'; auto with zfc.
Qed.

Theorem IN_small_small :
forall (E : Ens) (E' : Ens'),
IN E (inj E') -> EXType' _ (fun E1 : Ens' => EQ E (inj E1)).