Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  otiunsndisjX Structured version   Visualization version   GIF version

Theorem otiunsndisjX 41824
 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 399 . . . . 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 4676 . . . . . . . . . 10 (𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∃𝑐𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩})
4 simprl 811 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → 𝑎𝑉)
54adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → 𝑎𝑉)
6 simprl 811 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → 𝐵𝑋)
7 simpl 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → 𝑐𝑊)
8 otthg 5102 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝑉𝐵𝑋𝑐𝑊) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑𝐵 = 𝐵𝑐 = 𝑒)))
95, 6, 7, 8syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ ↔ (𝑎 = 𝑑𝐵 = 𝐵𝑐 = 𝑒)))
10 simp1 1131 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 = 𝑑𝐵 = 𝐵𝑐 = 𝑒) → 𝑎 = 𝑑)
119, 10syl6bi 243 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩ → 𝑎 = 𝑑))
1211con3d 148 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐𝑊 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
1312ex 449 . . . . . . . . . . . . . . . . . . . 20 (𝑐𝑊 → ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (¬ 𝑎 = 𝑑 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
1413com13 88 . . . . . . . . . . . . . . . . . . 19 𝑎 = 𝑑 → ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (𝑐𝑊 → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)))
1514imp31 447 . . . . . . . . . . . . . . . . . 18 (((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
1615adantr 472 . . . . . . . . . . . . . . . . 17 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
1716adantr 472 . . . . . . . . . . . . . . . 16 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒𝑊) → ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
18 velsn 4337 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} ↔ 𝑠 = ⟨𝑎, 𝐵, 𝑐⟩)
19 eqeq1 2764 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2019notbid 307 . . . . . . . . . . . . . . . . . . 19 (𝑠 = ⟨𝑎, 𝐵, 𝑐⟩ → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2118, 20sylbi 207 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2221adantl 473 . . . . . . . . . . . . . . . . 17 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2322adantr 472 . . . . . . . . . . . . . . . 16 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒𝑊) → (¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩ ↔ ¬ ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩))
2417, 23mpbird 247 . . . . . . . . . . . . . . 15 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒𝑊) → ¬ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
25 velsn 4337 . . . . . . . . . . . . . . 15 (𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩} ↔ 𝑠 = ⟨𝑑, 𝐵, 𝑒⟩)
2624, 25sylnibr 318 . . . . . . . . . . . . . 14 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) ∧ 𝑒𝑊) → ¬ 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
2726nrexdv 3139 . . . . . . . . . . . . 13 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ ∃𝑒𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
28 eliun 4676 . . . . . . . . . . . . 13 (𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩} ↔ ∃𝑒𝑊 𝑠 ∈ {⟨𝑑, 𝐵, 𝑒⟩})
2927, 28sylnibr 318 . . . . . . . . . . . 12 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) ∧ 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩}) → ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
3029ex 449 . . . . . . . . . . 11 (((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) ∧ 𝑐𝑊) → (𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩}))
3130rexlimdva 3169 . . . . . . . . . 10 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (∃𝑐𝑊 𝑠 ∈ {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩}))
323, 31syl5bi 232 . . . . . . . . 9 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} → ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩}))
3332ralrimiv 3103 . . . . . . . 8 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → ∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
34 oteq3 4564 . . . . . . . . . . . . 13 (𝑐 = 𝑒 → ⟨𝑑, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑒⟩)
3534sneqd 4333 . . . . . . . . . . . 12 (𝑐 = 𝑒 → {⟨𝑑, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑒⟩})
3635cbviunv 4711 . . . . . . . . . . 11 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩} = 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩}
3736eleq2i 2831 . . . . . . . . . 10 (𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
3837notbii 309 . . . . . . . . 9 𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
3938ralbii 3118 . . . . . . . 8 (∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩} ↔ ∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑒𝑊 {⟨𝑑, 𝐵, 𝑒⟩})
4033, 39sylibr 224 . . . . . . 7 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → ∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
41 disj 4160 . . . . . . 7 (( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅ ↔ ∀𝑠 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ¬ 𝑠 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
4240, 41sylibr 224 . . . . . 6 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)
4342olcd 407 . . . . 5 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉))) → (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
4443ex 449 . . . 4 𝑎 = 𝑑 → ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅)))
452, 44pm2.61i 176 . . 3 ((𝐵𝑋 ∧ (𝑎𝑉𝑑𝑉)) → (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
4645ralrimivva 3109 . 2 (𝐵𝑋 → ∀𝑎𝑉𝑑𝑉 (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
47 oteq1 4562 . . . . 5 (𝑎 = 𝑑 → ⟨𝑎, 𝐵, 𝑐⟩ = ⟨𝑑, 𝐵, 𝑐⟩)
4847sneqd 4333 . . . 4 (𝑎 = 𝑑 → {⟨𝑎, 𝐵, 𝑐⟩} = {⟨𝑑, 𝐵, 𝑐⟩})
4948iuneq2d 4699 . . 3 (𝑎 = 𝑑 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} = 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩})
5049disjor 4786 . 2 (Disj 𝑎𝑉 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ↔ ∀𝑎𝑉𝑑𝑉 (𝑎 = 𝑑 ∨ ( 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩} ∩ 𝑐𝑊 {⟨𝑑, 𝐵, 𝑐⟩}) = ∅))
5146, 50sylibr 224 1 (𝐵𝑋Disj 𝑎𝑉 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ∩ cin 3714  ∅c0 4058  {csn 4321  ⟨cotp 4329  ∪ ciun 4672  Disj wdisj 4772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rmo 3058  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330  df-iun 4674  df-disj 4773 This theorem is referenced by: (None)
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