Theorem otsndisj 5411

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	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑉 {⟨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.

Step	Hyp	Ref	Expression
1		otthg 5379	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑)))
2	1	3expa 1114	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑)))
3		simp3 1134	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑) → 𝑐 = 𝑑)
4	2, 3	syl6bi 255	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ → 𝑐 = 𝑑))
5	4	con3rr3 158	. . . . . . . . 9 ⊢ (¬ 𝑐 = 𝑑 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩))
6	5	imp 409	. . . . . . . 8 ⊢ ((¬ 𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉)) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
7	6	neqned 3025	. . . . . . 7 ⊢ ((¬ 𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉)) → ⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩)
8		disjsn2 4650	. . . . . . 7 ⊢ (⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩ → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)
9	7, 8	syl 17	. . . . . 6 ⊢ ((¬ 𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉)) → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅)
10	9	expcom 416	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (¬ 𝑐 = 𝑑 → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
11	10	orrd 859	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
12	11	adantrr 715	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
13	12	ralrimivva 3193	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
14		oteq3 4816	. . . 4 ⊢ (𝑐 = 𝑑 → ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
15	14	sneqd 4581	. . 3 ⊢ (𝑐 = 𝑑 → {⟨𝐴, 𝐵, 𝑐⟩} = {⟨𝐴, 𝐵, 𝑑⟩})
16	15	disjor 5048	. 2 ⊢ (Disj 𝑐 ∈ 𝑉 {⟨𝐴, 𝐵, 𝑐⟩} ↔ ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
17	13, 16	sylibr 236	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑉 {⟨𝐴, 𝐵, 𝑐⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140   ∩ cin 3937  ∅c0 4293  {csn 4569  ⟨cotp 4577  Disj wdisj 5033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rmo 3148  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-ot 4578  df-disj 5034

This theorem is referenced by: (None)