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Theorem enref 9025
Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
Hypothesis
Ref Expression
enref.1 𝐴 ∈ V
Assertion
Ref Expression
enref 𝐴𝐴

Proof of Theorem enref
StepHypRef Expression
1 enref.1 . 2 𝐴 ∈ V
2 enrefg 9024 . 2 (𝐴 ∈ V → 𝐴𝐴)
31, 2ax-mp 5 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3480   class class class wbr 5143  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986
This theorem is referenced by:  ener  9041  en0ALT  9059  pwen  9190  phplem2OLD  9255  phplem3OLD  9256  isinfOLD  9297  karden  9935  mappwen  10152  nnadju  10238  infmap2  10257  ackbij1lem5  10263  axcc4dom  10481  domtriomlem  10482  cfpwsdom  10624  0tsk  10795  fzennn  14009  qnnen  16249  rpnnen  16263  rexpen  16264  lmisfree  21862  met2ndci  24535  lgseisenlem2  27420  poimirlem9  37636  poimirlem26  37653  1aryenef  48566  2aryenef  48577
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