Theorem pwexg 4240

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 3629	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2276	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4239	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2838	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   _Vcvv 2776   ~Pcpw 3626

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  pwexd  4241  abssexg  4242  pwex  4243  snexg  4244  pwel  4280  uniexb  4538  xpexg  4807  fabexg  5485  mapex  6764  pmvalg  6769  fopwdom  6958  ssenen  6973  restid2  13195  toponsspwpwg  14609  tgdom  14659  distop  14672  epttop  14677  cldval  14686  ntrfval  14687  clsfval  14688  neifval  14727  neif  14728  neival  14730