Qed.

(* Specifications: *)
Theorem INC_tran : forall E E' E'' : Ens, INC E E' -> INC E' E'' -> INC E E''.
unfold INC in |- *; auto with zfc.
Theorem INC_sound_left :
forall E E' E'' : Ens, EQ E E' -> INC E E'' -> INC E' E''.
Theorem IN_Paire_right : forall E E' : Ens, IN E' (Paire E E').
unfold Paire in |- *; simpl in |- *; exists false; simpl in |- *;
auto with zfc.
Theorem Paire_IN :
forall E E' A : Ens, IN A (Paire E E') -> EQ A E \/ EQ A E'.
Theorem IN_Sing_EQ : forall E E' : Ens, IN E (Sing E') -> EQ E E'.
unfold Sing in |- *; intros E E' H; elim (Paire_IN E' E' E);
auto with zfc.
Theorem INC_Class_succ : forall E : Ens, INC E (Class_succ E).
unfold INC in |- *; unfold Class_succ in |- *.
Theorem INC_tran : forall E E' E'' : Ens, INC E E' -> INC E' E'' -> INC E E''.
unfold INC in |- *; auto with zfc.
Theorem INC_sound_left :
forall E E' E'' : Ens, EQ E E' -> INC E E'' -> INC E' E''.
Theorem IN_Paire_right : forall E E' : Ens, IN E' (Paire E E').
unfold Paire in |- *; simpl in |- *; exists false; simpl in |- *;
auto with zfc.
Theorem Paire_IN :
forall E E' A : Ens, IN A (Paire E E') -> EQ A E \/ EQ A E'.
Theorem IN_Sing_EQ : forall E E' : Ens, IN E (Sing E') -> EQ E E'.
unfold Sing in |- *; intros E E' H; elim (Paire_IN E' E' E);
auto with zfc.
Theorem INC_Succ : forall E : Ens, INC E (Succ E).
unfold INC in |- *; unfold Succ in |- *.
Theorem CINC_tran :
forall E E' E'' : Ens, CINC E E' -> CINC E' E'' -> CINC E E''.
Theorem CIN_Paire_right : forall E E' : Ens, CIN E' (Paire E E').
unfold Paire in |- *; simpl in |- *; exists false; simpl in |- *;
auto with zfc.
Theorem CIN_Sing_EQ : forall E E' : Ens, CIN E (Sing E') -> EQC E E'.
unfold Sing in |- *; intros E E' H; elim (Paire_CIN E' E' E);
auto with zfc.
Theorem EQC_EQ : forall E E' : Ens, EQC E E' -> EQ E E'.
simple induction E; intros A f ra; simple induction E'; intros B g rb;
simpl in |- *; simple induction 1; intros H1 H2; split.
Theorem IN_Big_small :
forall E : Ens, IN E Big -> EXType' _ (fun E' : Ens' => EQ E (inj E')).
Theorem CINC_tran :
forall E E' E'' : Ens, CINC E E' -> CINC E' E'' -> CINC E E''.
Theorem CIN_Paire_right : forall E E' : Ens, CIN E' (Paire E E').
unfold Paire in |- *; simpl in |- *; exists false; simpl in |- *;
auto with zfc.
Theorem CIN_Sing_EQ : forall E E' : Ens, CIN E (Sing E') -> EQC E E'.
unfold Sing in |- *; intros E E' H; elim (Paire_CIN E' E' E);
auto with zfc.
Theorem EQC_EQ : forall E E' : Ens, EQC E E' -> EQ E E'.
simple induction E; intros A f ra; simple induction E'; intros B g rb;
simpl in |- *; simple induction 1; intros H1 H2; split.
Theorem IN_Big_small :
forall E : Ens, IN E Big -> EXType' _ (fun E' : Ens' => EQ E (inj E')).