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Theorem otsndisj 5113
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 5082 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑)))
213expa 1111 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑)))
3 simp3 1132 . . . . . . . . . . 11 ((𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑) → 𝑐 = 𝑑)
42, 3syl6bi 243 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ → 𝑐 = 𝑑))
54con3rr3 152 . . . . . . . . 9 𝑐 = 𝑑 → (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩))
65imp 393 . . . . . . . 8 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
76neqned 2950 . . . . . . 7 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩)
8 disjsn2 4385 . . . . . . 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 398 . . . . 5 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (¬ 𝑐 = 𝑑 → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1110orrd 852 . . . 4 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1211adantrr 696 . . 3 (((𝐴𝑋𝐵𝑌) ∧ (𝑐𝑉𝑑𝑉)) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1312ralrimivva 3120 . 2 ((𝐴𝑋𝐵𝑌) → ∀𝑐𝑉𝑑𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
14 oteq3 4551 . . . 4 (𝑐 = 𝑑 → ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
1514sneqd 4329 . . 3 (𝑐 = 𝑑 → {⟨𝐴, 𝐵, 𝑐⟩} = {⟨𝐴, 𝐵, 𝑑⟩})
1615disjor 4769 . 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  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  cin 3722  c0 4063  {csn 4317  cotp 4325  Disj wdisj 4755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rmo 3069  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-ot 4326  df-disj 4756
This theorem is referenced by: (None)
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