Theorem otiunsndisj 4980

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

Assertion

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

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

Proof of Theorem otiunsndisj

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

Step	Hyp	Ref	Expression
1		eliun 4524	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ↔ ∃𝑐 ∈ (𝑊 ∖ {𝑎})𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩})
2		otthg 4954	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋 ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒)))
3		simp1 1061	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒) → 𝑎 = 𝑑)
4	2, 3	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋 ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ → 𝑎 = 𝑑))
5	4	con3d 148	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋 ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
6	5	3exp 1264	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ 𝑉 → (𝐵 ∈ 𝑋 → (𝑐 ∈ (𝑊 ∖ {𝑎}) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))))
7	6	impcom 446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉) → (𝑐 ∈ (𝑊 ∖ {𝑎}) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
8	7	com3r 87	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉) → (𝑐 ∈ (𝑊 ∖ {𝑎}) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
9	8	imp31 448	. . . . . . . . . . . . . . . . . 18 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
10		velsn 4193	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} ↔ 𝑠 = ⟨𝑎, 𝐵, 𝑐⟩)
11		eqeq1 2626	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
12	11	notbid 308	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
13	10, 12	sylbi 207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
14	9, 13	syl5ibrcom 237	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩))
15	14	imp 445	. . . . . . . . . . . . . . . 16 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
16		velsn 4193	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩} ↔ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
17	15, 16	sylnibr 319	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
18	17	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ (𝑊 ∖ {𝑑})) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
19	18	nrexdv 3001	. . . . . . . . . . . . 13 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ∃𝑒 ∈ (𝑊 ∖ {𝑑})𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
20		eliun 4524	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩} ↔ ∃𝑒 ∈ (𝑊 ∖ {𝑑})𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
21	19, 20	sylnibr 319	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
22	21	ex 450	. . . . . . . . . . 11 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}))
23	22	rexlimdva 3031	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → (∃𝑐 ∈ (𝑊 ∖ {𝑎})𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}))
24	1, 23	syl5bi 232	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → (𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}))
25	24	ralrimiv 2965	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
26		oteq3 4413	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑒 → ⟨𝑑, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
27	26	sneqd 4189	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → {⟨𝑑, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑒⟩})
28	27	cbviunv 4559	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} = ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}
29	28	eleq2i 2693	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} ↔ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
30	29	notbii 310	. . . . . . . . 9 ⊢ (¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} ↔ ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
31	30	ralbii 2980	. . . . . . . 8 ⊢ (∀𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
32	25, 31	sylibr 224	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩})
33		disj 4017	. . . . . . 7 ⊢ ((∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅ ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩})
34	32, 33	sylibr 224	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅)
35	34	expcom 451	. . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉) → (¬ 𝑎 = 𝑑 → (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
36	35	orrd 393	. . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
37	36	adantrr 753	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
38	37	ralrimivva 2971	. 2 ⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
39		sneq 4187	. . . . 5 ⊢ (𝑎 = 𝑑 → {𝑎} = {𝑑})
40	39	difeq2d 3728	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑊 ∖ {𝑎}) = (𝑊 ∖ {𝑑}))
41		oteq1 4411	. . . . 5 ⊢ (𝑎 = 𝑑 → ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑐⟩)
42	41	sneqd 4189	. . . 4 ⊢ (𝑎 = 𝑑 → {⟨𝑎, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑐⟩})
43	40, 42	iuneq12d 4546	. . 3 ⊢ (𝑎 = 𝑑 → ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} = ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩})
44	43	disjor 4634	. 2 ⊢ (Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ↔ ∀𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
45	38, 44	sylibr 224	1 ⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ∩ cin 3573  ∅c0 3915  {csn 4177  ⟨cotp 4185  ∪ ciun 4520  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-iun 4522  df-disj 4621

This theorem is referenced by: (None)