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Theorem pwexb 4287
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 4286 . 2  |-  ( ~P A  e.  _V  <->  U. ~P A  e.  _V )
2 unipw 4035 . . 3  |-  U. ~P A  =  A
32eleq1i 2153 . 2  |-  ( U. ~P A  e.  _V  <->  A  e.  _V )
41, 3bitr2i 183 1  |-  ( A  e.  _V  <->  ~P A  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1438   _Vcvv 2619   ~Pcpw 3425   U.cuni 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-uni 3649
This theorem is referenced by: (None)
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