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Theorem otsndisj 5529
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 5496 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑)))
213expa 1117 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑)))
3 simp3 1137 . . . . . . . . . . 11 ((𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑) → 𝑐 = 𝑑)
42, 3biimtrdi 253 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ → 𝑐 = 𝑑))
54con3rr3 155 . . . . . . . . 9 𝑐 = 𝑑 → (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩))
65imp 406 . . . . . . . 8 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
76neqned 2945 . . . . . . 7 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩)
8 disjsn2 4717 . . . . . . 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 413 . . . . 5 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (¬ 𝑐 = 𝑑 → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1110orrd 863 . . . 4 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1211adantrr 717 . . 3 (((𝐴𝑋𝐵𝑌) ∧ (𝑐𝑉𝑑𝑉)) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1312ralrimivva 3200 . 2 ((𝐴𝑋𝐵𝑌) → ∀𝑐𝑉𝑑𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
14 oteq3 4889 . . . 4 (𝑐 = 𝑑 → ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
1514sneqd 4643 . . 3 (𝑐 = 𝑑 → {⟨𝐴, 𝐵, 𝑐⟩} = {⟨𝐴, 𝐵, 𝑑⟩})
1615disjor 5130 . 2 (Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩} ↔ ∀𝑐𝑉𝑑𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1713, 16sylibr 234 1 ((𝐴𝑋𝐵𝑌) → Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  cin 3962  c0 4339  {csn 4631  cotp 4639  Disj wdisj 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rmo 3378  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-disj 5116
This theorem is referenced by: (None)
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