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Theorem pwexb 4564
Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexb  |-  ( A  e.  _V  <->  ~P A  e.  _V )

Proof of Theorem pwexb
StepHypRef Expression
1 uniexb 4563 . 2  |-  ( ~P A  e.  _V  <->  U. ~P A  e.  _V )
2 unipw 4224 . . 3  |-  U. ~P A  =  A
32eleq1i 2346 . 2  |-  ( U. ~P A  e.  _V  <->  A  e.  _V )
41, 3bitr2i 241 1  |-  ( A  e.  _V  <->  ~P A  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   U.cuni 3827
This theorem is referenced by:  pwuninel  6300  2pwuninel  7016  pwfi  7151  pwwf  7479  ranklim  7516  r1pw  7517  r1pwOLD  7518  isfin3  7922  isf34lem6  8006  isfin1-2  8011  pwfseqlem4  8284  pwfseqlem5  8285  gchpwdom  8296  hargch  8299  dis2ndc  17186  numufl  17610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828
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