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Theorem otiunsndisjX 46286
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 (𝐵𝑋Disj 𝑎𝑉 𝑐𝑊 {⟨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.
StepHypRef Expression
1 orc 864 . . . . 5 (𝑎 = 𝑑 → (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
21a1d 25 . . . 4 (𝑎 = 𝑑 → ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)))
3 eliun 5001 . . . . . . . . . 10 (𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∃𝑐𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩})
4 simprl 768 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → 𝑎𝑉)
54adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → 𝑎𝑉)
6 simprl 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → 𝐵𝑋)
7 simpl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → 𝑐𝑊)
8 otthg 5485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎𝑉𝐵𝑋𝑐𝑊) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑𝐵 = 𝐵𝑐 = 𝑒)))
95, 6, 7, 8syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑𝐵 = 𝐵𝑐 = 𝑒)))
10 simp1 1135 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 = 𝑑𝐵 = 𝐵𝑐 = 𝑒) → 𝑎 = 𝑑)
119, 10syl6bi 253 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ → 𝑎 = 𝑑))
1211con3d 152 . . . . . . . . . . . . . . . . . . . 20 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
1312ex 412 . . . . . . . . . . . . . . . . . . 19 (𝑐𝑊 → ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
1413com13 88 . . . . . . . . . . . . . . . . . 18 𝑎 = 𝑑 → ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (𝑐𝑊 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
1514imp31 417 . . . . . . . . . . . . . . . . 17 (((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
1615adantr 480 . . . . . . . . . . . . . . . 16 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
1716adantr 480 . . . . . . . . . . . . . . 15 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
18 velsn 4644 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} ↔ 𝑠 = ⟨𝑎, 𝐵, 𝑐⟩)
19 eqeq1 2735 . . . . . . . . . . . . . . . . . . 19 (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2019notbid 318 . . . . . . . . . . . . . . . . . 18 (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2118, 20sylbi 216 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2221adantl 481 . . . . . . . . . . . . . . . 16 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2322adantr 480 . . . . . . . . . . . . . . 15 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒𝑊) → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2417, 23mpbird 257 . . . . . . . . . . . . . 14 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒𝑊) → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
25 velsn 4644 . . . . . . . . . . . . . 14 (𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩} ↔ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
2624, 25sylnibr 329 . . . . . . . . . . . . 13 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒𝑊) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
2726nrexdv 3148 . . . . . . . . . . . 12 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ∃𝑒𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
28 eliun 5001 . . . . . . . . . . . 12 (𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩} ↔ ∃𝑒𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
2927, 28sylnibr 329 . . . . . . . . . . 11 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
3029rexlimdva2 3156 . . . . . . . . . 10 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (∃𝑐𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩}))
313, 30biimtrid 241 . . . . . . . . 9 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩}))
3231ralrimiv 3144 . . . . . . . 8 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → ∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
33 oteq3 4884 . . . . . . . . . . . . 13 (𝑐 = 𝑒 → ⟨𝑑, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
3433sneqd 4640 . . . . . . . . . . . 12 (𝑐 = 𝑒 → {⟨𝑑, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑒⟩})
3534cbviunv 5043 . . . . . . . . . . 11 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩} = 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩}
3635eleq2i 2824 . . . . . . . . . 10 (𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
3736notbii 320 . . . . . . . . 9 𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
3837ralbii 3092 . . . . . . . 8 (∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
3932, 38sylibr 233 . . . . . . 7 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → ∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
40 disj 4447 . . . . . . 7 (( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅ ↔ ∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
4139, 40sylibr 233 . . . . . 6 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)
4241olcd 871 . . . . 5 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
4342ex 412 . . . 4 𝑎 = 𝑑 → ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)))
442, 43pm2.61i 182 . . 3 ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
4544ralrimivva 3199 . 2 (𝐵𝑋 → ∀𝑎𝑉𝑑𝑉 (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
46 oteq1 4882 . . . . 5 (𝑎 = 𝑑 → ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑐⟩)
4746sneqd 4640 . . . 4 (𝑎 = 𝑑 → {⟨𝑎, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑐⟩})
4847iuneq2d 5026 . . 3 (𝑎 = 𝑑 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} = 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
4948disjor 5128 . 2 (Disj 𝑎𝑉 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∀𝑎𝑉𝑑𝑉 (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
5045, 49sylibr 233 1 (𝐵𝑋Disj 𝑎𝑉 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wrex 3069  cin 3947  c0 4322  {csn 4628  cotp 4636   ciun 4997  Disj wdisj 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-iun 4999  df-disj 5114
This theorem is referenced by: (None)
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