Theorem pwexg 4270

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 3655	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2300	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4269	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2864	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   ~Pcpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  pwexd  4271  abssexg  4272  pwex  4273  snexg  4274  pwel  4310  uniexb  4570  xpexg  4840  fabexg  5524  mapex  6822  pmvalg  6827  fopwdom  7021  ssenen  7036  restid2  13330  toponsspwpwg  14745  tgdom  14795  distop  14808  epttop  14813  cldval  14822  ntrfval  14823  clsfval  14824  neifval  14863  neif  14864  neival  14866