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Theorem pwexb 4755
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 4754 . 2  |-  ( ~P A  e.  _V  <->  U. ~P A  e.  _V )
2 unipw 4416 . . 3  |-  U. ~P A  =  A
32eleq1i 2501 . 2  |-  ( U. ~P A  e.  _V  <->  A  e.  _V )
41, 3bitr2i 243 1  |-  ( A  e.  _V  <->  ~P A  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1726   _Vcvv 2958   ~Pcpw 3801   U.cuni 4017
This theorem is referenced by:  pwuninel  6547  2pwuninel  7264  pwfi  7404  pwwf  7735  ranklim  7772  r1pw  7773  r1pwOLD  7774  isfin3  8178  isf34lem6  8262  isfin1-2  8267  pwfseqlem4  8539  pwfseqlem5  8540  gchpwdom  8551  hargch  8554  dis2ndc  17525  numufl  17949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-pr 3823  df-uni 4018
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