Theorem pwexg 4210

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 3605	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2262	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4209	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2821	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   ~Pcpw 3602

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167  df-pw 3604

This theorem is referenced by:  pwexd  4211  abssexg  4212  pwex  4213  snexg  4214  pwel  4248  uniexb  4505  xpexg  4774  fabexg  5442  mapex  6710  pmvalg  6715  fopwdom  6894  ssenen  6909  restid2  12862  toponsspwpwg  14201  tgdom  14251  distop  14264  epttop  14269  cldval  14278  ntrfval  14279  clsfval  14280  neifval  14319  neif  14320  neival  14322