Theorem pwexg 4298

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 3677	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2303	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4297	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2877	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   ~Pcpw 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676

This theorem is referenced by:  pwexd  4299  abssexg  4300  pwex  4301  snexg  4302  pwel  4339  uniexb  4599  xpexg  4869  fabexg  5559  mapex  6901  pmvalg  6906  fopwdom  7102  ssenen  7118  2omapfi  7284  restid2  13545  toponsspwpwg  15013  tgdom  15063  distop  15076  epttop  15081  cldval  15090  ntrfval  15091  clsfval  15092  neifval  15131  neif  15132  neival  15134