Theorem disjecxrn 38985

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 38979	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})
2		ecxrn 38979	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → [𝐵](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)})
3	1, 2	ineqan12d 4183	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)}))
4		inopab 5817	. . . . . . . . 9 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)}) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))}
5	3, 4	eqtrdi 2820	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))})
6		an4 668	. . . . . . . . 9 ⊢ (((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)) ↔ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
7	6	opabbii 5182	. . . . . . . 8 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))} = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))}
8	5, 7	eqtrdi 2820	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))})
9	8	neeq1d 3023	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))} ≠ ∅))
10		opabn0 5539	. . . . . 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 1982	. . . . 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 38926	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦)))
15		ecinn0 38926	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅ ↔ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
16	14, 15	anbi12d 643	. . . 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 3057	. . 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 3004	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964   ∩ cin 3912  ∅c0 4294   class class class wbr 5113  {copab 5177  [cec 8692   ⋉ cxrn 38747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7986  df-2nd 7987  df-ec 8696  df-xrn 38953

This theorem is referenced by:  disjecxrncnvep  38986