Theorem pwexg 4312

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 3688	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2307	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4311	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2883	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   ~Pcpw 3685

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  pwexd  4313  abssexg  4314  pwex  4315  snexg  4316  pwel  4353  uniexb  4614  xpexg  4884  fabexg  5574  mapex  6918  pmvalg  6923  fopwdom  7126  ssenen  7142  2omapfi  7310  restid2  13579  toponsspwpwg  15046  tgdom  15096  distop  15109  epttop  15114  cldval  15123  ntrfval  15124  clsfval  15125  neifval  15164  neif  15165  neival  15167