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

Theorem pwidg 3650
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 3210 . 2  |-  A  C_  A
2 elpwg 3645 . 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 1696    C_ wss 3165   ~Pcpw 3638
This theorem is referenced by:  pwid  3651  axpweq  4203  knatar  5873  brwdom2  7303  pwwf  7495  rankpwi  7511  canthp1lem2  8291  canthp1  8292  grothpw  8464  mremre  13522  submre  13523  baspartn  16708  fctop  16757  cctop  16759  ppttop  16760  epttop  16762  isopn3  16819  mretopd  16845  tsmsfbas  17826  esumcst  23451  pwsiga  23506  prsiga  23507  sigainb  23512  coinflipprob  23695  neibastop1  26411  neibastop2lem  26412  elrfi  26872
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