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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.
StepHypRef Expression
1 pweq 3387 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21eleq1d 2148 . 2  |-  ( x  =  A  ->  ( ~P x  e.  _V  <->  ~P A  e.  _V )
)
3 vex 2605 . . 3  |-  x  e. 
_V
43pwex 3955 . 2  |-  ~P x  e.  _V
52, 4vtoclg 2659 1  |-  ( A  e.  V  ->  ~P A  e.  _V )
Colors of variables: wff set class
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
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