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Theorem pwid 3804
Description: A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
pwid.1  |-  A  e. 
_V
Assertion
Ref Expression
pwid  |-  A  e. 
~P A

Proof of Theorem pwid
StepHypRef Expression
1 pwid.1 . 2  |-  A  e. 
_V
2 pwidg 3803 . 2  |-  ( A  e.  _V  ->  A  e.  ~P A )
31, 2ax-mp 8 1  |-  A  e. 
~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2948   ~Pcpw 3791
This theorem is referenced by:  r1ordg  7693  rankr1id  7777  cfss  8134  0ram  13376  bastg  17019  fincmp  17444  restlly  17534  ptbasfi  17601  zfbas  17916  ustfilxp  18230  metustfbasOLD  18583  metustfbas  18584  minveclem3b  19317  wilthlem3  20841  coinflipprob  24725  pwtrrVD  28792  mapdunirnN  32287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793
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