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Theorem elrelscnveq 36598
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
elrelscnveq (𝑅 ∈ Rels → (𝑅𝑅𝑅 = 𝑅))

Proof of Theorem elrelscnveq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrelscnveq3 36597 . . 3 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
2 cnvsym 6017 . . 3 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
31, 2bitr4di 289 . 2 (𝑅 ∈ Rels → (𝑅 = 𝑅𝑅𝑅))
4 eqcom 2747 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
53, 4bitr3di 286 1 (𝑅 ∈ Rels → (𝑅𝑅𝑅 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540   = wceq 1542  wcel 2110  wss 3892   class class class wbr 5079  ccnv 5588   Rels crels 36323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5595  df-rel 5596  df-cnv 5597  df-rels 36591
This theorem is referenced by:  elrelscnveq4  36600  dfsymrels4  36649
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