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Theorem pwexb 6382
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 6381 . 2  |-  ( ~P A  e.  _V  <->  U. ~P A  e.  _V )
2 unipw 4535 . . 3  |-  U. ~P A  =  A
32eleq1i 2500 . 2  |-  ( U. ~P A  e.  _V  <->  A  e.  _V )
41, 3bitr2i 250 1  |-  ( A  e.  _V  <->  ~P A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1756   _Vcvv 2966   ~Pcpw 3853   U.cuni 4084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rex 2715  df-v 2968  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-pw 3855  df-sn 3871  df-pr 3873  df-uni 4085
This theorem is referenced by:  pwuninel  6786  2pwuninel  7458  pwfi  7598  pwwf  8006  ranklim  8043  r1pw  8044  r1pwOLD  8045  isfin3  8457  isf34lem6  8541  isfin1-2  8546  pwfseqlem4  8821  pwfseqlem5  8822  gchpwdom  8829  hargch  8832  dis2ndc  19033  numufl  19457
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