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Theorem pwexb 6584
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 6583 . 2  |-  ( ~P A  e.  _V  <->  U. ~P A  e.  _V )
2 unipw 4687 . . 3  |-  U. ~P A  =  A
32eleq1i 2531 . 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 1823   _Vcvv 3106   ~Pcpw 3999   U.cuni 4235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001  df-sn 4017  df-pr 4019  df-uni 4236
This theorem is referenced by:  pwuninel  6996  2pwuninel  7665  pwfi  7807  pwwf  8216  ranklim  8253  r1pw  8254  r1pwALT  8255  isfin3  8667  isf34lem6  8751  isfin1-2  8756  pwfseqlem4  9029  pwfseqlem5  9030  gchpwdom  9037  hargch  9040  dis2ndc  20127  numufl  20582
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