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

Theorem pwidg 4028
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 3518 . 2  |-  A  C_  A
2 elpwg 4023 . 2  |-  ( A  e.  V  ->  ( A  e.  ~P A  <->  A 
C_  A ) )
31, 2mpbiri 233 1  |-  ( A  e.  V  ->  A  e.  ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    C_ wss 3471   ~Pcpw 4015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017
This theorem is referenced by:  pwid  4029  axpweq  4633  knatar  6254  brwdom2  8017  pwwf  8242  rankpwi  8258  canthp1lem2  9048  canthp1  9049  grothpw  9221  mremre  15021  submre  15022  baspartn  19582  fctop  19632  cctop  19634  ppttop  19635  epttop  19637  isopn3  19694  mretopd  19720  tsmsfbas  20752  gsumesum  28231  esumcst  28235  pwsiga  28303  prsiga  28304  sigainb  28309  neibastop1  30382  neibastop2lem  30383  elrfi  30831  dvnprodlem3  31948
  Copyright terms: Public domain W3C validator