Theorem pwexg 4263

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 3652	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2298	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4262	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2861	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   ~Pcpw 3649

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  pwexd  4264  abssexg  4265  pwex  4266  snexg  4267  pwel  4303  uniexb  4563  xpexg  4832  fabexg  5512  mapex  6799  pmvalg  6804  fopwdom  6993  ssenen  7008  restid2  13276  toponsspwpwg  14690  tgdom  14740  distop  14753  epttop  14758  cldval  14767  ntrfval  14768  clsfval  14769  neifval  14808  neif  14809  neival  14811