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Theorem elrelscnveq 34565
 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 eqcom 2767 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
2 elrelscnveq3 34564 . . 3 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
3 cnvsym 5668 . . 3 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
42, 3syl6bbr 278 . 2 (𝑅 ∈ Rels → (𝑅 = 𝑅𝑅𝑅))
51, 4syl5rbbr 275 1 (𝑅 ∈ Rels → (𝑅𝑅𝑅 = 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1630   = wceq 1632   ∈ wcel 2139   ⊆ wss 3715   class class class wbr 4804  ◡ccnv 5265   Rels crels 34298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-rels 34558 This theorem is referenced by:  elrelscnveq4  34567  dfsymrels4  34616
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