Theorem pwexg 3954

Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (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 3385	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2147	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vex 2604	. . 3 |- x e. _V
4	3	pwex 3953	. 2 |- ~P x e. _V
5	2, 4	vtoclg 2658	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   _Vcvv 2601   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  abssexg  3955  snexg  3956  pwel  3973  uniexb  4223  xpexg  4470  fabexg  5097  fopwdom  6333