Theorem pwexg 4195

Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)

Assertion

Ref	Expression
pwexg	|- ( A e. V -> ~P A e. _V )

Proof of Theorem pwexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 3593	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2258	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4194	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2812	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   _Vcvv 2752   ~Pcpw 3590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592

This theorem is referenced by:  pwexd  4196  abssexg  4197  pwex  4198  snexg  4199  pwel  4233  uniexb  4488  xpexg  4755  fabexg  5418  mapex  6672  pmvalg  6677  fopwdom  6854  ssenen  6869  restid2  12719  toponsspwpwg  13919  tgdom  13969  distop  13982  epttop  13987  cldval  13996  ntrfval  13997  clsfval  13998  neifval  14037  neif  14038  neival  14040