Theorem disjecxrn 38733

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 38727	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})
2		ecxrn 38727	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → [𝐵](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)})
3	1, 2	ineqan12d 4162	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)}))
4		inopab 5785	. . . . . . . . 9 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)}) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))}
5	3, 4	eqtrdi 2787	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))})
6		an4 657	. . . . . . . . 9 ⊢ (((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)) ↔ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
7	6	opabbii 5152	. . . . . . . 8 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))} = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))}
8	5, 7	eqtrdi 2787	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))})
9	8	neeq1d 2991	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))} ≠ ∅))
10		opabn0 5508	. . . . . 6 ⊢ ({⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))} ≠ ∅ ↔ ∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
11	9, 10	bitrdi 287	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ ∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
12		exdistrv 1957	. . . . 5 ⊢ (∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)) ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
13	11, 12	bitrdi 287	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
14		ecinn0 38674	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦)))
15		ecinn0 38674	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅ ↔ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
16	14, 15	anbi12d 633	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅) ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
17	13, 16	bitr4d 282	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅)))
18		neanior 3025	. . 3 ⊢ ((([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅) ↔ ¬ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅))
19	17, 18	bitrdi 287	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ ¬ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))
20	19	necon4abid 2972	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 206   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932   ∩ cin 3888  ∅c0 4273   class class class wbr 5085  {copab 5147  [cec 8641   ⋉ cxrn 38495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-1st 7942  df-2nd 7943  df-ec 8645  df-xrn 38701

This theorem is referenced by:  disjecxrncnvep  38734