Theorem pwexg 4181

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 3579	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2246	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4180	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2798	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2738   ~Pcpw 3576

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-pw 3578

This theorem is referenced by:  pwexd  4182  abssexg  4183  pwex  4184  snexg  4185  pwel  4219  uniexb  4474  xpexg  4741  fabexg  5404  mapex  6654  pmvalg  6659  fopwdom  6836  ssenen  6851  restid2  12697  toponsspwpwg  13525  tgdom  13575  distop  13588  epttop  13593  cldval  13602  ntrfval  13603  clsfval  13604  neifval  13643  neif  13644  neival  13646