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Theorem br2coss 36298
Description: Cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
Assertion
Ref Expression
br2coss ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))

Proof of Theorem br2coss
StepHypRef Expression
1 brcoss3 36293 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
2 cnvcosseq 36297 . . . . 5 𝑅 = ≀ 𝑅
32eceq2i 8432 . . . 4 [𝐴]𝑅 = [𝐴] ≀ 𝑅
42eceq2i 8432 . . . 4 [𝐵]𝑅 = [𝐵] ≀ 𝑅
53, 4ineq12i 4125 . . 3 ([𝐴]𝑅 ∩ [𝐵]𝑅) = ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅)
65neeq1i 3005 . 2 (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅)
71, 6bitrdi 290 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2110  wne 2940  cin 3865  c0 4237   class class class wbr 5053  ccnv 5550  [cec 8389  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-11 2158  ax-12 2175  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-ne 2941  df-ral 3066  df-rex 3067  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-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ec 8393  df-coss 36274
This theorem is referenced by: (None)
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