Theorem pwexg 3956

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 3387	. . 3 |- ( x = A -> ~P x = ~P A )
2	1	eleq1d 2148	. 2 |- ( x = A -> ( ~P x e. _V <-> ~P A e. _V ) )
3		vex 2605	. . 3 |- x e. _V
4	3	pwex 3955	. 2 |- ~P x e. _V
5	2, 4	vtoclg 2659	1 |- ( A e. V -> ~P A e. _V )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   _Vcvv 2602   ~Pcpw 3384

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3386

This theorem is referenced by:  abssexg  3957  snexg  3958  pwel  3975  uniexb  4225  xpexg  4474  fabexg  5102  fopwdom  6370