Theorem otsndisj 5524

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 5490	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑)))
2	1	3expa 1119	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑)))
3		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐵 ∧ 𝑐 = 𝑑) → 𝑐 = 𝑑)
4	2, 3	biimtrdi 253	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩ → 𝑐 = 𝑑))
5	4	con3rr3 155	. . . . . . . . 9 ⊢ (¬ 𝑐 = 𝑑 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩))
6	5	imp 406	. . . . . . . 8 ⊢ ((¬ 𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉)) → ¬ ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
7	6	neqned 2947	. . . . . . 7 ⊢ ((¬ 𝑐 = 𝑑 ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉)) → ⟨𝐴, 𝐵, 𝑐⟩ ≠ ⟨𝐴, 𝐵, 𝑑⟩)
8		disjsn2 4712	. . . . . . 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 413	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (¬ 𝑐 = 𝑑 → ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
11	10	orrd 864	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝑐 ∈ 𝑉) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
12	11	adantrr 717	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
13	12	ralrimivva 3202	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
14		oteq3 4884	. . . 4 ⊢ (𝑐 = 𝑑 → ⟨𝐴, 𝐵, 𝑐⟩ = ⟨𝐴, 𝐵, 𝑑⟩)
15	14	sneqd 4638	. . 3 ⊢ (𝑐 = 𝑑 → {⟨𝐴, 𝐵, 𝑐⟩} = {⟨𝐴, 𝐵, 𝑑⟩})
16	15	disjor 5125	. 2 ⊢ (Disj 𝑐 ∈ 𝑉 {⟨𝐴, 𝐵, 𝑐⟩} ↔ ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 = 𝑑 ∨ ({⟨𝐴, 𝐵, 𝑐⟩} ∩ {⟨𝐴, 𝐵, 𝑑⟩}) = ∅))
17	13, 16	sylibr 234	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑉 {⟨𝐴, 𝐵, 𝑐⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061   ∩ cin 3950  ∅c0 4333  {csn 4626  ⟨cotp 4634  Disj wdisj 5110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-disj 5111

This theorem is referenced by: (None)