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Theorem brcoss3 35558
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 5743 . . . . 5 ((𝐴𝑉𝑢 ∈ V) → (𝐴𝑅𝑢𝑢𝑅𝐴))
21elvd 3498 . . . 4 (𝐴𝑉 → (𝐴𝑅𝑢𝑢𝑅𝐴))
3 brcnvg 5743 . . . . 5 ((𝐵𝑊𝑢 ∈ V) → (𝐵𝑅𝑢𝑢𝑅𝐵))
43elvd 3498 . . . 4 (𝐵𝑊 → (𝐵𝑅𝑢𝑢𝑅𝐵))
52, 4bi2anan9 635 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝑢𝐵𝑅𝑢) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
65exbidv 1913 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴𝑅𝑢𝐵𝑅𝑢) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
7 ecinn0 35488 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑢(𝐴𝑅𝑢𝐵𝑅𝑢)))
8 brcoss 35556 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
96, 7, 83bitr4rd 313 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wex 1771  wcel 2105  wne 3013  Vcvv 3492  cin 3932  c0 4288   class class class wbr 5057  ccnv 5547  [cec 8276  ccoss 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280  df-coss 35539
This theorem is referenced by:  br2coss  35563  trcoss2  35604
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