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Theorem pwidg 4151
Description: Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
pwidg (𝐴𝑉𝐴 ∈ 𝒫 𝐴)

Proof of Theorem pwidg
StepHypRef Expression
1 ssid 3609 . 2 𝐴𝐴
2 elpwg 4144 . 2 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
31, 2mpbiri 248 1 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3560  𝒫 cpw 4136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-in 3567  df-ss 3574  df-pw 4138
This theorem is referenced by:  pwid  4152  axpweq  4812  knatar  6572  brwdom2  8438  pwwf  8630  rankpwi  8646  canthp1lem2  9435  canthp1  9436  grothpw  9608  mremre  16204  submre  16205  baspartn  20698  fctop  20748  cctop  20750  ppttop  20751  epttop  20753  isopn3  20810  mretopd  20836  tsmsfbas  21871  gsumesum  29944  esumcst  29948  pwsiga  30016  prsiga  30017  sigainb  30022  pwldsys  30043  ldgenpisyslem1  30049  carsggect  30203  neibastop1  32049  neibastop2lem  32050  topdifinfindis  32865  elrfi  36776  dssmapnvod  37835  ntrk0kbimka  37858  clsk3nimkb  37859  neik0pk1imk0  37866  ntrclscls00  37885  ntrneicls00  37908  pwssfi  38733  dvnprodlem3  39500  caragenunidm  40059
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