Theorem pwexg 4317

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 3691	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2307	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vpwex 4316	. 2 |- ~P x e. _V
4	2, 3	vtoclg 2883	1 |- ( A e. V -> ~P A e. _V )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   ~Pcpw 3688

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690

This theorem is used by:  pwexd  4318  abssexg  4319  pwex  4320  snexg  4321  pwel  4358  uniexb  4619  xpexg  4889  fabexg  5579  mapex  6928  pmvalg  6933  fopwdom  7136  ssenen  7152  2omapfi  7320  indv  9295  restid2  13602  toponsspwpwg  15123  tgdom  15173  distop  15186  epttop  15191  cldval  15200  ntrfval  15201  clsfval  15202  neifval  15241  neif  15242  neival  15244