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Theorem ecin0 35487
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 4397 . 2 (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅))
2 elecg 8321 . . . . . 6 ((𝑥 ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
32el2v1 35371 . . . . 5 (𝐴𝑉 → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
43adantr 481 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
5 elecALTV 35408 . . . . . . 7 ((𝐵𝑊𝑥 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
65elvd 3498 . . . . . 6 (𝐵𝑊 → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
76adantl 482 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
87notbid 319 . . . 4 ((𝐴𝑉𝐵𝑊) → (¬ 𝑥 ∈ [𝐵]𝑅 ↔ ¬ 𝐵𝑅𝑥))
94, 8imbi12d 346 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ (𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
109albidv 1912 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥(𝑥 ∈ [𝐴]𝑅 → ¬ 𝑥 ∈ [𝐵]𝑅) ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
111, 10syl5bb 284 1 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  c0 4288   class class class wbr 5057  [cec 8276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280
This theorem is referenced by:  ecinn0  35488
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