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Theorem symrelcoss3 36016
 Description: The class of cosets by 𝑅 is symmetric, see dfsymrel3 36097. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
Assertion
Ref Expression
symrelcoss3 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)

Proof of Theorem symrelcoss3
StepHypRef Expression
1 brcosscnvcoss 35990 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦𝑦𝑅𝑥))
21el2v 3449 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
32biimpi 219 . . 3 (𝑥𝑅𝑦𝑦𝑅𝑥)
43gen2 1798 . 2 𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)
5 relcoss 35979 . 2 Rel ≀ 𝑅
64, 5pm3.2i 474 1 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  Vcvv 3442   class class class wbr 5034  Rel wrel 5528   ≀ ccoss 35764 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-coss 35970 This theorem is referenced by:  symrelcoss2  36017  eqvrelcoss3  36164
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