Theorem disjecxrn 39487

Description: Two ways of saying that (𝑅 ⋉ 𝑆)-cosets are disjoint. (Contributed by Peter Mazsa, 19-Jun-2020.) (Revised by Peter Mazsa, 21-Aug-2023.)

Assertion

Ref	Expression
disjecxrn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ∅ ↔ (([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.

Step	Hyp	Ref	Expression
1		ecxrn 39481	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})
2		ecxrn 39481	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → [𝐵](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)})
3	1, 2	ineqan12d 4122	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)}))
4		inopab 5812	. . . . . . . . 9 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)}) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))}
5	3, 4	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))})
6		an4 669	. . . . . . . . 9 ⊢ (((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)) ↔ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
7	6	opabbii 5175	. . . . . . . 8 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))} = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))}
8	5, 7	eqtrdi 2814	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))})
9	8	neeq1d 3018	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))} ≠ ∅))
10		opabn0 5532	. . . . . 6 ⊢ ({⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))} ≠ ∅ ↔ ∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
11	9, 10	bitrdi 290	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ ∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
12		exdistrv 1989	. . . . 5 ⊢ (∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)) ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
13	11, 12	bitrdi 290	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
14		ecinn0 39428	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦)))
15		ecinn0 39428	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅ ↔ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
16	14, 15	anbi12d 644	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅) ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
17	13, 16	bitr4d 285	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅)))
18		neanior 3052	. . 3 ⊢ ((([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅) ↔ ¬ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅))
19	17, 18	bitrdi 290	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ ¬ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))
20	19	necon4abid 2999	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2959   ∩ cin 4092  ∅c0 4281   class class class wbr 5106  {copab 5170  [cec 8729   ⋉ cxrn 39249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fo 6553  df-fv 6555  df-1st 8011  df-2nd 8012  df-ec 8733  df-xrn 39455

This theorem is used by:  disjecxrncnvep  39488