Theorem pwexg 4141

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 3546	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2226	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4140	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2772	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   _Vcvv 2712   ~Pcpw 3543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545

This theorem is referenced by:  pwexd  4142  abssexg  4143  pwex  4144  snexg  4145  pwel  4178  uniexb  4432  xpexg  4699  fabexg  5356  mapex  6596  pmvalg  6601  fopwdom  6778  ssenen  6793  restid2  12331  toponsspwpwg  12391  tgdom  12443  distop  12456  epttop  12461  cldval  12470  ntrfval  12471  clsfval  12472  neifval  12511  neif  12512  neival  12514