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Theorem elrelscnveq 35891
 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 35890 . . 3 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
2 cnvsym 5945 . . 3 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
31, 2syl6bbr 292 . 2 (𝑅 ∈ Rels → (𝑅 = 𝑅𝑅𝑅))
4 eqcom 2808 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
53, 4bitr3di 289 1 (𝑅 ∈ Rels → (𝑅𝑅𝑅 = 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2112   ⊆ wss 3884   class class class wbr 5033  ◡ccnv 5522   Rels crels 35614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-rels 35884 This theorem is referenced by:  elrelscnveq4  35893  dfsymrels4  35942
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