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Theorem brcoss3 34729
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 5534 . . . . 5 ((𝐴𝑉𝑢 ∈ V) → (𝐴𝑅𝑢𝑢𝑅𝐴))
21elvd 3419 . . . 4 (𝐴𝑉 → (𝐴𝑅𝑢𝑢𝑅𝐴))
3 brcnvg 5534 . . . . 5 ((𝐵𝑊𝑢 ∈ V) → (𝐵𝑅𝑢𝑢𝑅𝐵))
43elvd 3419 . . . 4 (𝐵𝑊 → (𝐵𝑅𝑢𝑢𝑅𝐵))
52, 4bi2anan9 629 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝑢𝐵𝑅𝑢) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
65exbidv 2020 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴𝑅𝑢𝐵𝑅𝑢) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
7 ecinn0 34659 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑢(𝐴𝑅𝑢𝐵𝑅𝑢)))
8 brcoss 34727 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
96, 7, 83bitr4rd 304 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wex 1878  wcel 2164  wne 2999  Vcvv 3414  cin 3797  c0 4144   class class class wbr 4873  ccnv 5341  [cec 8007  ccoss 34517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-cnv 5350  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ec 8011  df-coss 34710
This theorem is referenced by:  br2coss  34734  trcoss2  34775
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