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Theorem disjecxrn 36783
Description: Two ways of saying that (𝑅𝑆)-cosets are disjoint. (Contributed by Peter Mazsa, 19-Jun-2020.) (Revised by Peter Mazsa, 21-Aug-2023.)
Assertion
Ref Expression
disjecxrn ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))

Proof of Theorem disjecxrn
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecxrn 36781 . . . . . . . . . 10 (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)})
2 ecxrn 36781 . . . . . . . . . 10 (𝐵𝑊 → [𝐵](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦𝐵𝑆𝑧)})
31, 2ineqan12d 4173 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦𝐵𝑆𝑧)}))
4 inopab 5784 . . . . . . . . 9 ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦𝐵𝑆𝑧)}) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦𝐵𝑆𝑧))}
53, 4eqtrdi 2794 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦𝐵𝑆𝑧))})
6 an4 655 . . . . . . . . 9 (((𝐴𝑅𝑦𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦𝐵𝑆𝑧)) ↔ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧)))
76opabbii 5171 . . . . . . . 8 {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦𝐵𝑆𝑧))} = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))}
85, 7eqtrdi 2794 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))})
98neeq1d 3002 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))} ≠ ∅))
10 opabn0 5509 . . . . . 6 ({⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))} ≠ ∅ ↔ ∃𝑦𝑧((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧)))
119, 10bitrdi 287 . . . . 5 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ ∃𝑦𝑧((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))))
12 exdistrv 1960 . . . . 5 (∃𝑦𝑧((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧)) ↔ (∃𝑦(𝐴𝑅𝑦𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧𝐵𝑆𝑧)))
1311, 12bitrdi 287 . . . 4 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ (∃𝑦(𝐴𝑅𝑦𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧𝐵𝑆𝑧))))
14 ecinn0 36746 . . . . 5 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑦(𝐴𝑅𝑦𝐵𝑅𝑦)))
15 ecinn0 36746 . . . . 5 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅ ↔ ∃𝑧(𝐴𝑆𝑧𝐵𝑆𝑧)))
1614, 15anbi12d 632 . . . 4 ((𝐴𝑉𝐵𝑊) → ((([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅) ↔ (∃𝑦(𝐴𝑅𝑦𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧𝐵𝑆𝑧))))
1713, 16bitr4d 282 . . 3 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅)))
18 neanior 3036 . . 3 ((([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅) ↔ ¬ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅))
1917, 18bitrdi 287 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ ¬ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))
2019necon4abid 2983 1 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  wne 2942  cin 3908  c0 4281   class class class wbr 5104  {copab 5166  [cec 8605  cxrn 36565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fo 6500  df-fv 6502  df-1st 7914  df-2nd 7915  df-ec 8609  df-xrn 36765
This theorem is referenced by:  disjecxrncnvep  36784
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