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Theorem pwid 3638
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 3637 . 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 1684   _Vcvv 2788   ~Pcpw 3625
This theorem is referenced by:  r1ordg  7450  rankr1id  7534  cfss  7891  0ram  13067  bastg  16704  fincmp  17120  restlly  17209  ptbasfi  17276  zfbas  17591  minveclem3b  18792  wilthlem3  20308  nZdef  25180  pwtrrVD  28600  pwtrrOLD  28601  mapdunirnN  31840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627
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