Theorem otiunsndisjX 42175

Description: The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)

Assertion

Ref	Expression
otiunsndisjX	⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩})

Distinct variable groups:   𝐵,𝑎,𝑐   𝑉,𝑎,𝑐   𝑊,𝑎,𝑐   𝑋,𝑎,𝑐

Proof of Theorem otiunsndisjX

Dummy variables 𝑑 𝑒 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 898	. . . . 5 ⊢ (𝑎 = 𝑑 → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
2	1	a1d 25	. . . 4 ⊢ (𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)))
3		eliun 4744	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∃𝑐 ∈ 𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩})
4		simprl 787	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
5	4	adantl 475	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝑎 ∈ 𝑉)
6		simprl 787	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝐵 ∈ 𝑋)
7		simpl 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝑐 ∈ 𝑊)
8		otthg 5174	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋 ∧ 𝑐 ∈ 𝑊) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒)))
9	5, 6, 7, 8	syl3anc 1494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒)))
10		simp1 1170	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒) → 𝑎 = 𝑑)
11	9, 10	syl6bi 245	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ → 𝑎 = 𝑑))
12	11	con3d 150	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
13	12	ex 403	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∈ 𝑊 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
14	13	com13 88	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐 ∈ 𝑊 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
15	14	imp31 410	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
16	15	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
17	16	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
18		velsn 4413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} ↔ 𝑠 = ⟨𝑎, 𝐵, 𝑐⟩)
19		eqeq1 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
20	19	notbid 310	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
21	18, 20	sylbi 209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
22	21	adantl 475	. . . . . . . . . . . . . . . 16 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
23	22	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
24	17, 23	mpbird 249	. . . . . . . . . . . . . 14 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
25		velsn 4413	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩} ↔ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
26	24, 25	sylnibr 321	. . . . . . . . . . . . 13 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
27	26	nrexdv 3209	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ∃𝑒 ∈ 𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
28		eliun 4744	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩} ↔ ∃𝑒 ∈ 𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
29	27, 28	sylnibr 321	. . . . . . . . . . 11 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
30	29	rexlimdva2 3243	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (∃𝑐 ∈ 𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}))
31	3, 30	syl5bi 234	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}))
32	31	ralrimiv 3174	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ∀𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
33		oteq3 4634	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑒 → ⟨𝑑, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
34	33	sneqd 4409	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → {⟨𝑑, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑒⟩})
35	34	cbviunv 4779	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} = ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}
36	35	eleq2i 2898	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
37	36	notbii 312	. . . . . . . . 9 ⊢ (¬ 𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
38	37	ralbii 3189	. . . . . . . 8 ⊢ (∀𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
39	32, 38	sylibr 226	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ∀𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
40		disj 4241	. . . . . . 7 ⊢ ((∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅ ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
41	39, 40	sylibr 226	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)
42	41	olcd 905	. . . . 5 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
43	42	ex 403	. . . 4 ⊢ (¬ 𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)))
44	2, 43	pm2.61i 177	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
45	44	ralrimivva 3180	. 2 ⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
46		oteq1 4632	. . . . 5 ⊢ (𝑎 = 𝑑 → ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑐⟩)
47	46	sneqd 4409	. . . 4 ⊢ (𝑎 = 𝑑 → {⟨𝑎, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑐⟩})
48	47	iuneq2d 4767	. . 3 ⊢ (𝑎 = 𝑑 → ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} = ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
49	48	disjor 4855	. 2 ⊢ (Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∀𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
50	45, 49	sylibr 226	1 ⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 878   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118   ∩ cin 3797  ∅c0 4144  {csn 4397  ⟨cotp 4405  ∪ ciun 4740  Disj wdisj 4841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rmo 3125  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-ot 4406  df-iun 4742  df-disj 4842

This theorem is referenced by: (None)