Theorem pwexg 4224

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 3619	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2274	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4223	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2833	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   ~Pcpw 3616

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618

This theorem is referenced by:  pwexd  4225  abssexg  4226  pwex  4227  snexg  4228  pwel  4262  uniexb  4520  xpexg  4789  fabexg  5463  mapex  6741  pmvalg  6746  fopwdom  6933  ssenen  6948  restid2  13080  toponsspwpwg  14494  tgdom  14544  distop  14557  epttop  14562  cldval  14571  ntrfval  14572  clsfval  14573  neifval  14612  neif  14613  neival  14615