Theorem pwexg 4293

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 3672	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2301	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4292	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2875	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   ~Pcpw 3669

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-pw 3671

This theorem is referenced by:  pwexd  4294  abssexg  4295  pwex  4296  snexg  4297  pwel  4334  uniexb  4594  xpexg  4864  fabexg  5554  mapex  6888  pmvalg  6893  fopwdom  7089  ssenen  7105  2omapfi  7271  restid2  13461  toponsspwpwg  14887  tgdom  14937  distop  14950  epttop  14955  cldval  14964  ntrfval  14965  clsfval  14966  neifval  15005  neif  15006  neival  15008