Theorem disjecxrn 38779

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 38773	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})
2		ecxrn 38773	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → [𝐵](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)})
3	1, 2	ineqan12d 4151	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)}))
4		inopab 5772	. . . . . . . . 9 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ∩ {⟨𝑦, 𝑧⟩ ∣ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)}) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))}
5	3, 4	eqtrdi 2790	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))})
6		an4 662	. . . . . . . . 9 ⊢ (((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)) ↔ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
7	6	opabbii 5139	. . . . . . . 8 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ∧ (𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))} = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))}
8	5, 7	eqtrdi 2790	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))})
9	8	neeq1d 2993	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ {⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))} ≠ ∅))
10		opabn0 5495	. . . . . 6 ⊢ ({⟨𝑦, 𝑧⟩ ∣ ((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))} ≠ ∅ ↔ ∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
11	9, 10	bitrdi 288	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ ∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
12		exdistrv 1962	. . . . 5 ⊢ (∃𝑦∃𝑧((𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ (𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)) ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
13	11, 12	bitrdi 288	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
14		ecinn0 38720	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦)))
15		ecinn0 38720	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅ ↔ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧)))
16	14, 15	anbi12d 638	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅) ↔ (∃𝑦(𝐴𝑅𝑦 ∧ 𝐵𝑅𝑦) ∧ ∃𝑧(𝐴𝑆𝑧 ∧ 𝐵𝑆𝑧))))
17	13, 16	bitr4d 283	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅)))
18		neanior 3027	. . 3 ⊢ ((([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ∧ ([𝐴]𝑆 ∩ [𝐵]𝑆) ≠ ∅) ↔ ¬ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅))
19	17, 18	bitrdi 288	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ ¬ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))
20	19	necon4abid 2974	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 207   ∧ wa 396   ∨ wo 853   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934   ∩ cin 3882  ∅c0 4261   class class class wbr 5072  {copab 5134  [cec 8631   ⋉ cxrn 38541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493  df-1st 7931  df-2nd 7932  df-ec 8635  df-xrn 38747

This theorem is referenced by:  disjecxrncnvep  38780