Theorem pwexg 4112

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 3518	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2209	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4111	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2749	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   _Vcvv 2689   ~Pcpw 3515

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517

This theorem is referenced by:  pwexd  4113  abssexg  4114  pwex  4115  snexg  4116  pwel  4148  uniexb  4402  xpexg  4661  fabexg  5318  mapex  6556  pmvalg  6561  fopwdom  6738  ssenen  6753  restid2  12168  toponsspwpwg  12228  tgdom  12280  distop  12293  epttop  12298  cldval  12307  ntrfval  12308  clsfval  12309  neifval  12348  neif  12349  neival  12351