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Theorem symrelcoss3 35147
Description: The class of cosets by 𝑅 is symmetric, see dfsymrel3 35228. (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 35121 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦𝑦𝑅𝑥))
21el2v 3423 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
32biimpi 208 . . 3 (𝑥𝑅𝑦𝑦𝑅𝑥)
43gen2 1759 . 2 𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)
5 relcoss 35110 . 2 Rel ≀ 𝑅
64, 5pm3.2i 463 1 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1505  Vcvv 3416   class class class wbr 4929  Rel wrel 5412  ccoss 34894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-coss 35101
This theorem is referenced by:  symrelcoss2  35148  eqvrelcoss3  35295
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