Theorem pwexg 4319

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 3691	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2307	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4318	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2883	1 |- ( A e. V -> ~P A e. _V )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   ~Pcpw 3688

This proof depends on 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 4251  ax-pow 4313

This proof 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 3690

This theorem is used by:  pwexd  4320  abssexg  4321  pwex  4322  snexg  4323  pwel  4360  uniexb  4621  xpexg  4891  fabexg  5582  mapex  6932  pmvalg  6937  fopwdom  7140  ssenen  7156  2omapfi  7326  indv  9304  restid2  13661  toponsspwpwg  15248  tgdom  15298  distop  15311  epttop  15316  cldval  15325  ntrfval  15326  clsfval  15327  neifval  15366  neif  15367  neival  15369