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Theorem ider 8685
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 = 𝑦𝑥 = 𝑦)
2 df-id 5532 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
31, 2eqer 8684 1 I Er V
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3446   I cid 5531   Er wer 8646
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-er 8649
This theorem is referenced by: (None)
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