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Theorem pwexb 4365
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 4364 . 2  |-  ( ~P A  e.  _V  <->  U. ~P A  e.  _V )
2 unipw 4109 . . 3  |-  U. ~P A  =  A
32eleq1i 2183 . 2  |-  ( U. ~P A  e.  _V  <->  A  e.  _V )
41, 3bitr2i 184 1  |-  ( A  e.  _V  <->  ~P A  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1465   _Vcvv 2660   ~Pcpw 3480   U.cuni 3706
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-uni 3707
This theorem is referenced by: (None)
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