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Theorem brcoss3 38434
Description: Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.)
Assertion
Ref Expression
brcoss3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))

Proof of Theorem brcoss3
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 brcnvg 5890 . . . . 5 ((𝐴𝑉𝑢 ∈ V) → (𝐴𝑅𝑢𝑢𝑅𝐴))
21elvd 3486 . . . 4 (𝐴𝑉 → (𝐴𝑅𝑢𝑢𝑅𝐴))
3 brcnvg 5890 . . . . 5 ((𝐵𝑊𝑢 ∈ V) → (𝐵𝑅𝑢𝑢𝑅𝐵))
43elvd 3486 . . . 4 (𝐵𝑊 → (𝐵𝑅𝑢𝑢𝑅𝐵))
52, 4bi2anan9 638 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝑢𝐵𝑅𝑢) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
65exbidv 1921 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴𝑅𝑢𝐵𝑅𝑢) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
7 ecinn0 38354 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑢(𝐴𝑅𝑢𝐵𝑅𝑢)))
8 brcoss 38432 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
96, 7, 83bitr4rd 312 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1779  wcel 2108  wne 2940  Vcvv 3480  cin 3950  c0 4333   class class class wbr 5143  ccnv 5684  [cec 8743  ccoss 38182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-coss 38412
This theorem is referenced by:  br2coss  38439  trcoss2  38485
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