Theorem pwexg 3933

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 3362	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2106	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vex 2560	. . 3 |- x e. _V
4	3	pwex 3932	. 2 |- ~P x e. _V
5	2, 4	vtoclg 2613	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   _Vcvv 2557   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  abssexg  3934  snexgOLD  3935  snexg  3936  pwel  3954  uniexb  4205  xpexg  4452  fabexg  5077  fopwdom  6310