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Theorem pwexg 4072
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 3481 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21eleq1d 2184 . 2  |-  ( x  =  A  ->  ( ~P x  e.  _V  <->  ~P A  e.  _V )
)
3 vpwex 4071 . 2  |-  ~P x  e.  _V
42, 3vtoclg 2718 1  |-  ( A  e.  V  ->  ~P A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   _Vcvv 2658   ~Pcpw 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480
This theorem is referenced by:  pwexd  4073  abssexg  4074  pwex  4075  snexg  4076  pwel  4108  uniexb  4362  xpexg  4621  fabexg  5278  mapex  6514  pmvalg  6519  fopwdom  6696  ssenen  6711  restid2  12035  toponsspwpwg  12095  tgdom  12147  distop  12160  epttop  12165  cldval  12174  ntrfval  12175  clsfval  12176  neifval  12215  neif  12216  neival  12218
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