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Theorem br2coss 34516
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 34511 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
2 cnvcosseq 34515 . . . . 5 𝑅 = ≀ 𝑅
32eceq2i 34364 . . . 4 [𝐴]𝑅 = [𝐴] ≀ 𝑅
42eceq2i 34364 . . . 4 [𝐵]𝑅 = [𝐵] ≀ 𝑅
53, 4ineq12i 3955 . . 3 ([𝐴]𝑅 ∩ [𝐵]𝑅) = ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅)
65neeq1i 2996 . 2 (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅)
71, 6syl6bb 276 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  wne 2932  cin 3714  c0 4058   class class class wbr 4804  ccnv 5265  [cec 7909  ccoss 34296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ec 7913  df-coss 34492
This theorem is referenced by: (None)
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