Theorem pwexg 4213

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 3608	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2265	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4212	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2824	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   ~Pcpw 3605

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607

This theorem is referenced by:  pwexd  4214  abssexg  4215  pwex  4216  snexg  4217  pwel  4251  uniexb  4508  xpexg  4777  fabexg  5445  mapex  6713  pmvalg  6718  fopwdom  6897  ssenen  6912  restid2  12919  toponsspwpwg  14258  tgdom  14308  distop  14321  epttop  14326  cldval  14335  ntrfval  14336  clsfval  14337  neifval  14376  neif  14377  neival  14379