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Theorem pwid 3579
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 3578 . 2  |-  ( A  e.  _V  ->  A  e.  ~P A )
31, 2ax-mp 10 1  |-  A  e. 
~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1621   _Vcvv 2740   ~Pcpw 3566
This theorem is referenced by:  r1ordg  7383  rankr1id  7467  cfss  7824  0ram  12994  bastg  16631  fincmp  17047  restlly  17136  ptbasfi  17203  zfbas  17518  minveclem3b  18719  wilthlem3  20235  nZdef  24512  pwtrrVD  27613  pwtrrOLD  27614  mapdunirnN  30970
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-in 3101  df-ss 3108  df-pw 3568
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