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Theorem ecin0 38354
Description: Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have no elements in common. (Contributed by Peter Mazsa, 1-Dec-2018.)
Assertion
Ref Expression
ecin0 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊

Proof of Theorem ecin0
StepHypRef Expression
1 disj1 4451 . 2 (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅))
2 elecg 8790 . . . . . 6 ((𝑥 ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
32el2v1 38225 . . . . 5 (𝐴𝑉 → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
43adantr 480 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
5 elecALTV 38268 . . . . . . 7 ((𝐵𝑊𝑥 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
65elvd 3485 . . . . . 6 (𝐵𝑊 → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
76adantl 481 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
87notbid 318 . . . 4 ((𝐴𝑉𝐵𝑊) → (¬ 𝑥 ∈ [𝐵]𝑅 ↔ ¬ 𝐵𝑅𝑥))
94, 8imbi12d 344 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ (𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
109albidv 1919 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
111, 10bitrid 283 1 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wcel 2107  Vcvv 3479  cin 3949  c0 4332   class class class wbr 5142  [cec 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748
This theorem is referenced by:  ecinn0  38355
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