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Theorem enref 8395
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 8394 . 2 (𝐴 ∈ V → 𝐴𝐴)
31, 2ax-mp 5 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2081  Vcvv 3437   class class class wbr 4966  cen 8359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-en 8363
This theorem is referenced by:  ener  8409  en0  8425  pwen  8542  phplem2  8549  phplem3  8550  isinf  8582  pssnn  8587  karden  9175  mappwen  9389  infmap2  9491  ackbij1lem5  9497  axcc4dom  9714  domtriomlem  9715  cfpwsdom  9857  0tsk  10028  fzennn  13191  qnnen  15404  rpnnen  15418  rexpen  15419  lmisfree  20673  met2ndci  22820  lgseisenlem2  25639  poimirlem9  34457  poimirlem26  34474
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