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Theorem pwid 3542
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 3541 . 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 2727   ~Pcpw 3530
This theorem is referenced by:  r1ordg  7334  rankr1id  7418  cfss  7775  0ram  12941  bastg  16536  fincmp  16952  restlly  17041  ptbasfi  17108  zfbas  17423  minveclem3b  18624  wilthlem3  20140  nZdef  24346  pwtrrVD  27290  pwtrrOLD  27291  mapdunirnN  30529
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-in 3085  df-ss 3089  df-pw 3532
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