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Theorem otiunsndisj 5529
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 (𝐵𝑋Disj 𝑎𝑉 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.
StepHypRef Expression
1 eliun 4999 . . . . . . . . . 10 (𝑠 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ↔ ∃𝑐 ∈ (𝑊 ∖ {𝑎})𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩})
2 otthg 5495 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎𝑉𝐵𝑋𝑐 ∈ (𝑊 ∖ {𝑎})) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑𝐵 = 𝐵𝑐 = 𝑒)))
3 simp1 1135 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 = 𝑑𝐵 = 𝐵𝑐 = 𝑒) → 𝑎 = 𝑑)
42, 3biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑉𝐵𝑋𝑐 ∈ (𝑊 ∖ {𝑎})) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ → 𝑎 = 𝑑))
54con3d 152 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎𝑉𝐵𝑋𝑐 ∈ (𝑊 ∖ {𝑎})) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
653exp 1118 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑉 → (𝐵𝑋 → (𝑐 ∈ (𝑊 ∖ {𝑎}) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))))
76impcom 407 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑋𝑎𝑉) → (𝑐 ∈ (𝑊 ∖ {𝑎}) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
87com3r 87 . . . . . . . . . . . . . . . . . 18 𝑎 = 𝑑 → ((𝐵𝑋𝑎𝑉) → (𝑐 ∈ (𝑊 ∖ {𝑎}) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
98imp31 417 . . . . . . . . . . . . . . . . 17 (((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
10 velsn 4646 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} ↔ 𝑠 = ⟨𝑎, 𝐵, 𝑐⟩)
11 eqeq1 2738 . . . . . . . . . . . . . . . . . . 19 (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
1211notbid 318 . . . . . . . . . . . . . . . . . 18 (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
1310, 12sylbi 217 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
149, 13syl5ibrcom 247 . . . . . . . . . . . . . . . 16 (((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) → (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩))
1514imp 406 . . . . . . . . . . . . . . 15 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
16 velsn 4646 . . . . . . . . . . . . . . 15 (𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩} ↔ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
1715, 16sylnibr 329 . . . . . . . . . . . . . 14 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
1817adantr 480 . . . . . . . . . . . . 13 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒 ∈ (𝑊 ∖ {𝑑})) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
1918nrexdv 3146 . . . . . . . . . . . 12 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ∃𝑒 ∈ (𝑊 ∖ {𝑑})𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
20 eliun 4999 . . . . . . . . . . . 12 (𝑠 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩} ↔ ∃𝑒 ∈ (𝑊 ∖ {𝑑})𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
2119, 20sylnibr 329 . . . . . . . . . . 11 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) ∧ 𝑐 ∈ (𝑊 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
2221rexlimdva2 3154 . . . . . . . . . 10 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) → (∃𝑐 ∈ (𝑊 ∖ {𝑎})𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}))
231, 22biimtrid 242 . . . . . . . . 9 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) → (𝑠 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}))
2423ralrimiv 3142 . . . . . . . 8 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) → ∀𝑠 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
25 oteq3 4888 . . . . . . . . . . . . 13 (𝑐 = 𝑒 → ⟨𝑑, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
2625sneqd 4642 . . . . . . . . . . . 12 (𝑐 = 𝑒 → {⟨𝑑, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑒⟩})
2726cbviunv 5044 . . . . . . . . . . 11 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} = 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩}
2827eleq2i 2830 . . . . . . . . . 10 (𝑠 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} ↔ 𝑠 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
2928notbii 320 . . . . . . . . 9 𝑠 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩} ↔ ¬ 𝑠 𝑒 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑒⟩})
3029ralbii 3090 . . . . . . . 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⟩})
3124, 30sylibr 234 . . . . . . 7 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) → ∀𝑠 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩})
32 disj 4455 . . . . . . 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⟩})
3331, 32sylibr 234 . . . . . 6 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋𝑎𝑉)) → ( 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅)
3433expcom 413 . . . . 5 ((𝐵𝑋𝑎𝑉) → (¬ 𝑎 = 𝑑 → ( 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
3534orrd 863 . . . 4 ((𝐵𝑋𝑎𝑉) → (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
3635adantrr 717 . . 3 ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
3736ralrimivva 3199 . 2 (𝐵𝑋 → ∀𝑎𝑉𝑑𝑉 (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
38 sneq 4640 . . . 4 (𝑎 = 𝑑 → {𝑎} = {𝑑})
3938difeq2d 4135 . . 3 (𝑎 = 𝑑 → (𝑊 ∖ {𝑎}) = (𝑊 ∖ {𝑑}))
40 oteq1 4886 . . . 4 (𝑎 = 𝑑 → ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑐⟩)
4140sneqd 4642 . . 3 (𝑎 = 𝑑 → {⟨𝑎, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑐⟩})
4239, 41disjiunb 5137 . 2 (Disj 𝑎𝑉 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ↔ ∀𝑎𝑉𝑑𝑉 (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐 ∈ (𝑊 ∖ {𝑑}){⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
4337, 42sylibr 234 1 (𝐵𝑋Disj 𝑎𝑉 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  cdif 3959  cin 3961  c0 4338  {csn 4630  cotp 4638   ciun 4995  Disj wdisj 5114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rmo 3377  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-iun 4997  df-disj 5115
This theorem is referenced by: (None)
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