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Theorem symrelcoss3 37799
Description: The class of cosets by 𝑅 is symmetric, see dfsymrel3 37884. (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 37768 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦𝑦𝑅𝑥))
21el2v 3481 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
32biimpi 215 . . 3 (𝑥𝑅𝑦𝑦𝑅𝑥)
43gen2 1797 . 2 𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)
5 relcoss 37757 . 2 Rel ≀ 𝑅
64, 5pm3.2i 470 1 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1538  Vcvv 3473   class class class wbr 5148  Rel wrel 5681  ccoss 37507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-coss 37745
This theorem is referenced by:  symrelcoss2  37800  eqvrelcoss3  37952
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