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Theorem pwidg 3637
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 3197 . 2  |-  A  C_  A
2 elpwg 3632 . 2  |-  ( A  e.  V  ->  ( A  e.  ~P A  <->  A 
C_  A ) )
31, 2mpbiri 224 1  |-  ( A  e.  V  ->  A  e.  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    C_ wss 3152   ~Pcpw 3625
This theorem is referenced by:  pwid  3638  axpweq  4187  knatar  5857  brwdom2  7287  pwwf  7479  rankpwi  7495  canthp1lem2  8275  canthp1  8276  grothpw  8448  mremre  13506  submre  13507  baspartn  16692  fctop  16741  cctop  16743  ppttop  16744  epttop  16746  isopn3  16803  mretopd  16829  tsmsfbas  17810  esumcst  23436  pwsiga  23491  prsiga  23492  sigainb  23497  coinflipprob  23680  neibastop1  26308  neibastop2lem  26309  elrfi  26769
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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