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Theorem ecin0 36744
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 4408 . 2 (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅))
2 elecg 8649 . . . . . 6 ((𝑥 ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
32el2v1 36607 . . . . 5 (𝐴𝑉 → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
43adantr 481 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
5 elecALTV 36657 . . . . . . 7 ((𝐵𝑊𝑥 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
65elvd 3450 . . . . . 6 (𝐵𝑊 → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
76adantl 482 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
87notbid 317 . . . 4 ((𝐴𝑉𝐵𝑊) → (¬ 𝑥 ∈ [𝐵]𝑅 ↔ ¬ 𝐵𝑅𝑥))
94, 8imbi12d 344 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ (𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
109albidv 1923 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
111, 10bitrid 282 1 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  Vcvv 3443  cin 3907  c0 4280   class class class wbr 5103  [cec 8604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8608
This theorem is referenced by:  ecinn0  36745
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