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Theorem pwexg 4015
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.
StepHypRef Expression
1 pweq 3432 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21eleq1d 2156 . 2  |-  ( x  =  A  ->  ( ~P x  e.  _V  <->  ~P A  e.  _V )
)
3 vpwex 4014 . 2  |-  ~P x  e.  _V
42, 3vtoclg 2679 1  |-  ( A  e.  V  ->  ~P A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619   ~Pcpw 3429
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-pw 3431
This theorem is referenced by:  pwexd  4016  abssexg  4017  pwex  4018  snexg  4019  pwel  4045  uniexb  4295  xpexg  4552  fabexg  5198  mapex  6409  pmvalg  6414  fopwdom  6550  ssenen  6565  toponsspwpwg  11573
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