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Theorem pwidg 4584
Description: A set is an element of its 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 3970 . 2 𝐴𝐴
2 elpwg 4567 . 2 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
31, 2mpbiri 258 1 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wss 3914  𝒫 cpw 4564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566
This theorem is referenced by:  pwidb  4585  pwid  4586  axpweq  5309  knatar  7306  brwdom2  9517  pwwf  9751  rankpwi  9767  canthp1lem2  10597  canthp1  10598  mremre  17492  submre  17493  baspartn  22327  fctop  22377  cctop  22379  ppttop  22380  epttop  22382  isopn3  22440  mretopd  22466  tsmsfbas  23502  gsumesum  32722  esumcst  32726  pwsiga  32793  prsiga  32794  sigainb  32799  pwldsys  32820  ldgenpisyslem1  32826  carsggect  32982  ex-sategoelel  34079  neibastop1  34884  neibastop2lem  34885  topdifinfindis  35867  elrfi  41064  dssmapnvod  42384  ntrk0kbimka  42403  clsk3nimkb  42404  neik0pk1imk0  42411  ntrclscls00  42430  ntrneicls00  42453  pwssfi  43345  dvnprodlem3  44279  caragenunidm  44839
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