Theorem otiunsndisj 5461

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 4925	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ↔ ∃𝑐 ∈ (𝑊 ∖ {𝑎})𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩})
2		otthg 5425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋 ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒)))
3		simp1 1142	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 = 𝑑 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑒) → 𝑎 = 𝑑)
4	2, 3	biimtrdi 254	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋 ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ → 𝑎 = 𝑑))
5	4	con3d 152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋 ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
6	5	3exp 1125	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ 𝑉 → (𝐵 ∈ 𝑋 → (𝑐 ∈ (𝑊 ∖ {𝑎}) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))))
7	6	impcom 408	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉) → (𝑐 ∈ (𝑊 ∖ {𝑎}) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
8	7	com3r 87	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉) → (𝑐 ∈ (𝑊 ∖ {𝑎}) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
9	8	imp31 418	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
10		velsn 4571	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} ↔ 𝑠 = ⟨𝑎, 𝐵, 𝑐⟩)
11		eqeq1 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
12	11	notbid 319	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
13	10, 12	sylbi 218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
14	9, 13	syl5ibrcom 248	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩))
15	14	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
16		velsn 4571	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩} ↔ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
17	15, 16	sylnibr 330	. . . . . . . . . . . . . 14 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
18	17	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ (𝑊 ∖ {𝑑})) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
19	18	nrexdv 3134	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ∃𝑒 ∈ (𝑊 ∖ {𝑑})𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
20		eliun 4925	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩} ↔ ∃𝑒 ∈ (𝑊 ∖ {𝑑})𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
21	19, 20	sylnibr 330	. . . . . . . . . . 11 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
22	21	rexlimdva2 3142	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → (∃𝑐 ∈ (𝑊 ∖ {𝑎})𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}))
23	1, 22	biimtrid 243	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → (𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}))
24	23	ralrimiv 3130	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
25		oteq3 4815	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑒 → ⟨𝑑, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
26	25	sneqd 4567	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → {⟨𝑑, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑒⟩})
27	26	cbviunv 4968	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} = ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}
28	27	eleq2i 2831	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} ↔ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
29	28	notbii 321	. . . . . . . . 9 ⊢ (¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} ↔ ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
30	29	ralbii 3085	. . . . . . . 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⟩})
31	24, 30	sylibr 235	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩})
32		disj 4378	. . . . . . 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⟩})
33	31, 32	sylibr 235	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉)) → (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅)
34	33	expcom 414	. . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉) → (¬ 𝑎 = 𝑑 → (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
35	34	orrd 869	. . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑎 ∈ 𝑉) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
36	35	adantrr 723	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
37	36	ralrimivva 3182	. 2 ⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ ∪ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
38		sneq 4565	. . . 4 ⊢ (𝑎 = 𝑑 → {𝑎} = {𝑑})
39	38	difeq2d 4057	. . 3 ⊢ (𝑎 = 𝑑 → (𝑊 ∖ {𝑎}) = (𝑊 ∖ {𝑑}))
40		oteq1 4813	. . . 4 ⊢ (𝑎 = 𝑑 → ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑐⟩)
41	40	sneqd 4567	. . 3 ⊢ (𝑎 = 𝑑 → {⟨𝑎, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑐⟩})
42	39, 41	disjiunb 5062	. 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⟩}) = ∅))
43	37, 42	sylibr 235	1 ⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∖ cdif 3880   ∩ cin 3882  ∅c0 4261  {csn 4555  ⟨cotp 4563  ∪ ciun 4921  Disj wdisj 5039

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-11 2168  ax-ext 2711  ax-sep 5218  ax-pr 5362

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-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rmo 3344  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-ot 4564  df-iun 4923  df-disj 5040

This theorem is referenced by: (None)