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Theorem ider 7337
Description: The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ider  |-  _I  Er  _V

Proof of Theorem ider
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2  |-  ( x  =  y  ->  x  =  y )
2 dfid3 4785 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
31, 2eqer 7336 1  |-  _I  Er  _V
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3106    _I cid 4779    Er wer 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-er 7303
This theorem is referenced by: (None)
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