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Theorem otsndisj 5396
 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 5364 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑)))
213expa 1115 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑)))
3 simp3 1135 . . . . . . . . . . 11 ((𝐴 = 𝐴𝐵 = 𝐵𝑐 = 𝑑) → 𝑐 = 𝑑)
42, 3syl6bi 256 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ → 𝑐 = 𝑑))
54con3rr3 158 . . . . . . . . 9 𝑐 = 𝑑 → (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩))
65imp 410 . . . . . . . 8 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
76neqned 3021 . . . . . . 7 ((¬ 𝑐 = 𝑑 ∧ ((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉)) → ⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩)
8 disjsn2 4633 . . . . . . 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 417 . . . . 5 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (¬ 𝑐 = 𝑑 → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1110orrd 860 . . . 4 (((𝐴𝑋𝐵𝑌) ∧ 𝑐𝑉) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1211adantrr 716 . . 3 (((𝐴𝑋𝐵𝑌) ∧ (𝑐𝑉𝑑𝑉)) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1312ralrimivva 3186 . 2 ((𝐴𝑋𝐵𝑌) → ∀𝑐𝑉𝑑𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
14 oteq3 4800 . . . 4 (𝑐 = 𝑑 → ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
1514sneqd 4562 . . 3 (𝑐 = 𝑑 → {⟨𝐴, 𝐵, 𝑐⟩} = {⟨𝐴, 𝐵, 𝑑⟩})
1615disjor 5032 . 2 (Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩} ↔ ∀𝑐𝑉𝑑𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
1713, 16sylibr 237 1 ((𝐴𝑋𝐵𝑌) → Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133   ∩ cin 3918  ∅c0 4276  {csn 4550  ⟨cotp 4558  Disj wdisj 5017 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rmo 3141  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-ot 4559  df-disj 5018 This theorem is referenced by: (None)
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