Theorem pwexg 4104

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 3513	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2208	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4103	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2746	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   _Vcvv 2686   ~Pcpw 3510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512

This theorem is referenced by:  pwexd  4105  abssexg  4106  pwex  4107  snexg  4108  pwel  4140  uniexb  4394  xpexg  4653  fabexg  5310  mapex  6548  pmvalg  6553  fopwdom  6730  ssenen  6745  restid2  12129  toponsspwpwg  12189  tgdom  12241  distop  12254  epttop  12259  cldval  12268  ntrfval  12269  clsfval  12270  neifval  12309  neif  12310  neival  12312