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Theorem symrelcoss3 36320
Description: The class of cosets by 𝑅 is symmetric, see dfsymrel3 36401. (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 36294 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦𝑦𝑅𝑥))
21el2v 3416 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
32biimpi 219 . . 3 (𝑥𝑅𝑦𝑦𝑅𝑥)
43gen2 1804 . 2 𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)
5 relcoss 36283 . 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 1541  Vcvv 3408   class class class wbr 5053  Rel wrel 5556  ccoss 36070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-coss 36274
This theorem is referenced by:  symrelcoss2  36321  eqvrelcoss3  36468
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