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Theorem ecinn0 38354
Description: Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have some elements in common. (Contributed by Peter Mazsa, 23-Jan-2019.)
Assertion
Ref Expression
ecinn0 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊

Proof of Theorem ecinn0
StepHypRef Expression
1 ecin0 38353 . . 3 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
21necon3abid 2977 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ¬ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
3 notnotb 315 . . . . 5 (𝐵𝑅𝑥 ↔ ¬ ¬ 𝐵𝑅𝑥)
43anbi2i 623 . . . 4 ((𝐴𝑅𝑥𝐵𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ ¬ ¬ 𝐵𝑅𝑥))
54exbii 1848 . . 3 (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ ¬ ¬ 𝐵𝑅𝑥))
6 exanali 1859 . . 3 (∃𝑥(𝐴𝑅𝑥 ∧ ¬ ¬ 𝐵𝑅𝑥) ↔ ¬ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥))
75, 6bitri 275 . 2 (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ↔ ¬ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥))
82, 7bitr4di 289 1 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1538  wex 1779  wcel 2108  wne 2940  cin 3950  c0 4333   class class class wbr 5143  [cec 8743
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
This theorem is referenced by:  disjecxrn  38390  brcoss3  38434  brcosscnv2  38474
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