Theorem grothpwex 9649

Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 9645. Note that ax-pow 4843 is not used by the proof. Use axpweq 4842 to obtain ax-pow 4843. Use pwex 4848 or pwexg 4850 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
grothpwex	|- ~P x e. _V

Proof of Theorem grothpwex

Dummy variables y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . 7 |- ( ( ~P z C_ y /\ E. w e. y ~P z C_ w ) -> ~P z C_ y )
2	1	ralimi 2952	. . . . . 6 |- ( A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) -> A. z e. y ~P z C_ y )
3		pweq 4161	. . . . . . . 8 |- ( z = x -> ~P z = ~P x )
4	3	sseq1d 3632	. . . . . . 7 |- ( z = x -> ( ~P z C_ y <-> ~P x C_ y ) )
5	4	rspccv 3306	. . . . . 6 |- ( A. z e. y ~P z C_ y -> ( x e. y -> ~P x C_ y ) )
6	2, 5	syl 17	. . . . 5 |- ( A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) -> ( x e. y -> ~P x C_ y ) )
7	6	anim2i 593	. . . 4 |- ( ( x e. y /\ A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) ) -> ( x e. y /\ ( x e. y -> ~P x C_ y ) ) )
8	7	3adant3 1081	. . 3 |- ( ( x e. y /\ A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) /\ A. z e. ~P y ( z ~~ y \/ z e. y ) ) -> ( x e. y /\ ( x e. y -> ~P x C_ y ) ) )
9		pm3.35 611	. . 3 |- ( ( x e. y /\ ( x e. y -> ~P x C_ y ) ) -> ~P x C_ y )
10		vex 3203	. . . 4 |- y e. _V
11	10	ssex 4802	. . 3 |- ( ~P x C_ y -> ~P x e. _V )
12	8, 9, 11	3syl 18	. 2 |- ( ( x e. y /\ A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) /\ A. z e. ~P y ( z ~~ y \/ z e. y ) ) -> ~P x e. _V )
13		axgroth5 9646	. 2 |- E. y ( x e. y /\ A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) /\ A. z e. ~P y ( z ~~ y \/ z e. y ) )
14	12, 13	exlimiiv 1859	1 |- ~P x e. _V

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-groth 9645

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  isrnsigaOLD  30175