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Theorem ecin0 34658
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 4243 . 2 (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅))
2 elecg 8050 . . . . . 6 ((𝑥 ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
32el2v1 34540 . . . . 5 (𝐴𝑉 → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
43adantr 474 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
5 elecALTV 34577 . . . . . . 7 ((𝐵𝑊𝑥 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
65elvd 3419 . . . . . 6 (𝐵𝑊 → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
76adantl 475 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
87notbid 310 . . . 4 ((𝐴𝑉𝐵𝑊) → (¬ 𝑥 ∈ [𝐵]𝑅 ↔ ¬ 𝐵𝑅𝑥))
94, 8imbi12d 336 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ (𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
109albidv 2019 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
111, 10syl5bb 275 1 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wal 1654   = wceq 1656  wcel 2164  Vcvv 3414  cin 3797  c0 4144   class class class wbr 4873  [cec 8007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-cnv 5350  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ec 8011
This theorem is referenced by:  ecinn0  34659
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