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Theorem enref 7586
Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
Hypothesis
Ref Expression
enref.1  |-  A  e. 
_V
Assertion
Ref Expression
enref  |-  A  ~~  A

Proof of Theorem enref
StepHypRef Expression
1 enref.1 . 2  |-  A  e. 
_V
2 enrefg 7585 . 2  |-  ( A  e.  _V  ->  A  ~~  A )
31, 2ax-mp 5 1  |-  A  ~~  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1842   _Vcvv 3059   class class class wbr 4395    ~~ cen 7551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-en 7555
This theorem is referenced by:  ener  7600  en0  7616  pwen  7728  phplem2  7735  phplem3  7736  isinf  7768  pssnn  7773  karden  8345  mappwen  8525  cdacomen  8593  infmap2  8630  ackbij1lem5  8636  axcc4dom  8853  domtriomlem  8854  cfpwsdom  8991  0tsk  9163  fzennn  12119  qnnen  14156  rpnnen  14169  rexpen  14170  lmisfree  19169  met2ndci  21317  lgseisenlem2  24006
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