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Theorem otsndisj 4939
Description: The singletons consisting of ordered triples which have distinct third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Assertion
Ref Expression
otsndisj ((𝐴𝑋𝐵𝑌) → Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩})
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐   𝑉,𝑐   𝑋,𝑐   𝑌,𝑐

Proof of Theorem otsndisj
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 otthg 4914 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑)))
213expa 1262 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑)))
3 simp3 1061 . . . . . . . . . . 11 ((𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑) → 𝑐 = 𝑑)
42, 3syl6bi 243 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ → 𝑐 = 𝑑))
54con3rr3 151 . . . . . . . . 9 𝑐 = 𝑑 → (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩))
65imp 445 . . . . . . . 8 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
76neqned 2797 . . . . . . 7 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩)
8 disjsn2 4217 . . . . . . 7 (⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩ → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)
97, 8syl 17 . . . . . 6 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)
109expcom 451 . . . . 5 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (¬ 𝑐 = 𝑑 → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1110orrd 393 . . . 4 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1211adantrr 752 . . 3 (((𝐴𝑋𝐵𝑌) ∧ (𝑐𝑉𝑑𝑉)) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1312ralrimivva 2965 . 2 ((𝐴𝑋𝐵𝑌) → ∀𝑐𝑉𝑑𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
14 oteq3 4381 . . . 4 (𝑐 = 𝑑 → ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
1514sneqd 4160 . . 3 (𝑐 = 𝑑 → {⟨𝐴, 𝐵, 𝑐⟩} = {⟨𝐴, 𝐵, 𝑑⟩})
1615disjor 4597 . 2 (Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩} ↔ ∀𝑐𝑉𝑑𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1713, 16sylibr 224 1 ((𝐴𝑋𝐵𝑌) → Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  cin 3554  c0 3891  {csn 4148  cotp 4156  Disj wdisj 4583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rmo 2915  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-ot 4157  df-disj 4584
This theorem is referenced by: (None)
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