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Theorem brcoss3 35710
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 5722 . . . . 5 ((𝐴𝑉𝑢 ∈ V) → (𝐴𝑅𝑢𝑢𝑅𝐴))
21elvd 3476 . . . 4 (𝐴𝑉 → (𝐴𝑅𝑢𝑢𝑅𝐴))
3 brcnvg 5722 . . . . 5 ((𝐵𝑊𝑢 ∈ V) → (𝐵𝑅𝑢𝑢𝑅𝐵))
43elvd 3476 . . . 4 (𝐵𝑊 → (𝐵𝑅𝑢𝑢𝑅𝐵))
52, 4bi2anan9 637 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝑢𝐵𝑅𝑢) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
65exbidv 1922 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴𝑅𝑢𝐵𝑅𝑢) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
7 ecinn0 35639 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑢(𝐴𝑅𝑢𝐵𝑅𝑢)))
8 brcoss 35708 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
96, 7, 83bitr4rd 314 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wex 1780  wcel 2114  wne 3006  Vcvv 3470  cin 3908  c0 4265   class class class wbr 5038  ccnv 5526  [cec 8261  ccoss 35485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5039  df-opab 5101  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ec 8265  df-coss 35691
This theorem is referenced by:  br2coss  35715  trcoss2  35756
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