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Theorem pwidg 3811
Description: Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
pwidg  |-  ( A  e.  V  ->  A  e.  ~P A )

Proof of Theorem pwidg
StepHypRef Expression
1 ssid 3367 . 2  |-  A  C_  A
2 elpwg 3806 . 2  |-  ( A  e.  V  ->  ( A  e.  ~P A  <->  A 
C_  A ) )
31, 2mpbiri 225 1  |-  ( A  e.  V  ->  A  e.  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    C_ wss 3320   ~Pcpw 3799
This theorem is referenced by:  pwid  3812  axpweq  4376  knatar  6080  brwdom2  7541  pwwf  7733  rankpwi  7749  canthp1lem2  8528  canthp1  8529  grothpw  8701  mremre  13829  submre  13830  baspartn  17019  fctop  17068  cctop  17070  ppttop  17071  epttop  17073  isopn3  17130  mretopd  17156  tsmsfbas  18157  gsumesum  24451  esumcst  24455  pwsiga  24513  prsiga  24514  sigainb  24519  neibastop1  26388  neibastop2lem  26389  elrfi  26748
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801
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