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

Theorem pwid 3651
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 3650 . 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 1696   _Vcvv 2801   ~Pcpw 3638
This theorem is referenced by:  r1ordg  7466  rankr1id  7550  cfss  7907  0ram  13083  bastg  16720  fincmp  17136  restlly  17225  ptbasfi  17292  zfbas  17607  minveclem3b  18808  wilthlem3  20324  nZdef  25283  pwtrrVD  28916  pwtrrOLD  28917  mapdunirnN  32462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640
  Copyright terms: Public domain W3C validator