MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwexb Structured version   Unicode version

Theorem pwexb 6582
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 6581 . 2  |-  ( ~P A  e.  _V  <->  U. ~P A  e.  _V )
2 unipw 4690 . . 3  |-  U. ~P A  =  A
32eleq1i 2537 . 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 1762   _Vcvv 3106   ~Pcpw 4003   U.cuni 4238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rex 2813  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-pw 4005  df-sn 4021  df-pr 4023  df-uni 4239
This theorem is referenced by:  pwuninel  6994  2pwuninel  7662  pwfi  7804  pwwf  8214  ranklim  8251  r1pw  8252  r1pwOLD  8253  isfin3  8665  isf34lem6  8749  isfin1-2  8754  pwfseqlem4  9029  pwfseqlem5  9030  gchpwdom  9037  hargch  9040  dis2ndc  19720  numufl  20144
  Copyright terms: Public domain W3C validator