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Theorem disjecxrn 38371
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 38369 . . . . . . . . . 10 (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)})
2 ecxrn 38369 . . . . . . . . . 10 (𝐵𝑊 → [𝐵](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦𝐵𝑆𝑧)})
31, 2ineqan12d 4230 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦𝐵𝑆𝑧)}))
4 inopab 5842 . . . . . . . . 9 ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦𝐵𝑆𝑧)}) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦𝐵𝑆𝑧))}
53, 4eqtrdi 2791 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦𝐵𝑆𝑧))})
6 an4 656 . . . . . . . . 9 (((𝐴𝑅𝑦𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦𝐵𝑆𝑧)) ↔ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧)))
76opabbii 5215 . . . . . . . 8 {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦𝐵𝑆𝑧))} = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))}
85, 7eqtrdi 2791 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))})
98neeq1d 2998 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))} ≠ ∅))
10 opabn0 5563 . . . . . 6 ({⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))} ≠ ∅ ↔ ∃𝑦𝑧((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧)))
119, 10bitrdi 287 . . . . 5 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ ∃𝑦𝑧((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧))))
12 exdistrv 1953 . . . . 5 (∃𝑦𝑧((𝐴𝑅𝑦𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧𝐵𝑆𝑧)) ↔ (∃𝑦(𝐴𝑅𝑦𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧𝐵𝑆𝑧)))
1311, 12bitrdi 287 . . . 4 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ (∃𝑦(𝐴𝑅𝑦𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧𝐵𝑆𝑧))))
14 ecinn0 38335 . . . . 5 ((𝐴𝑉𝐵𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑦(𝐴𝑅𝑦𝐵𝑅𝑦)))
15 ecinn0 38335 . . . . 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 3033 . . 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 2979 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 206  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  wne 2938  cin 3962  c0 4339   class class class wbr 5148  {copab 5210  [cec 8742  cxrn 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-ec 8746  df-xrn 38353
This theorem is referenced by:  disjecxrncnvep  38372
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