Theorem otiunsndisjX 44658

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 863	. . . . 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 4925	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∃𝑐 ∈ 𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩})
4		simprl 767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
5	4	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝑎 ∈ 𝑉)
6		simprl 767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝐵 ∈ 𝑋)
7		simpl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → 𝑐 ∈ 𝑊)
8		otthg 5394	. . . . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒)))
10		simp1 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒) → 𝑎 = 𝑑)
11	9, 10	syl6bi 252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ → 𝑎 = 𝑑))
12	11	con3d 152	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐 ∈ 𝑊 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
13	12	ex 412	. . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
16	15	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
17	16	adantr 480	. . . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} ↔ 𝑠 = ⟨𝑎, 𝐵, 𝑐⟩)
19		eqeq1 2742	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
20	19	notbid 317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
21	18, 20	sylbi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
22	21	adantl 481	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . 14 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
25		velsn 4574	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩} ↔ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
26	24, 25	sylnibr 328	. . . . . . . . . . . . 13 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ 𝑊) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
27	26	nrexdv 3197	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ∃𝑒 ∈ 𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
28		eliun 4925	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩} ↔ ∃𝑒 ∈ 𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
29	27, 28	sylnibr 328	. . . . . . . . . . 11 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) ∧ 𝑐 ∈ 𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
30	29	rexlimdva2 3215	. . . . . . . . . 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 241	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}))
32	31	ralrimiv 3106	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ∀𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
33		oteq3 4812	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑒 → ⟨𝑑, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
34	33	sneqd 4570	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → {⟨𝑑, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑒⟩})
35	34	cbviunv 4966	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} = ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩}
36	35	eleq2i 2830	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
37	36	notbii 319	. . . . . . . . 9 ⊢ (¬ 𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ¬ 𝑠 ∈ ∪ 𝑒 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
38	37	ralbii 3090	. . . . . . . 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 233	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ∀𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
40		disj 4378	. . . . . . 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 233	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)
42	41	olcd 870	. . . . 5 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
43	42	ex 412	. . . 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 182	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
45	44	ralrimivva 3114	. 2 ⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
46		oteq1 4810	. . . . 5 ⊢ (𝑎 = 𝑑 → ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑐⟩)
47	46	sneqd 4570	. . . 4 ⊢ (𝑎 = 𝑑 → {⟨𝑎, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑐⟩})
48	47	iuneq2d 4950	. . 3 ⊢ (𝑎 = 𝑑 → ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩} = ∪ 𝑐 ∈ 𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
49	48	disjor 5050	. 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 233	1 ⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {⟨𝑎, 𝐵, 𝑐⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882  ∅c0 4253  {csn 4558  ⟨cotp 4566  ∪ ciun 4921  Disj wdisj 5035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rmo 3071  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-ot 4567  df-iun 4923  df-disj 5036

This theorem is referenced by: (None)