Theorem otthne 5426

Description: Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 31-Jan-2025.)

Hypotheses

Ref	Expression
otthne.1	⊢ 𝐴 ∈ V
otthne.2	⊢ 𝐵 ∈ V
otthne.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
otthne	⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹))

Proof of Theorem otthne

Step	Hyp	Ref	Expression
1		otthne.1	. . . . 5 ⊢ 𝐴 ∈ V
2		otthne.2	. . . . 5 ⊢ 𝐵 ∈ V
3		otthne.3	. . . . 5 ⊢ 𝐶 ∈ V
4	1, 2, 3	otth 5424	. . . 4 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))
5	4	notbii 321	. . 3 ⊢ (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))
6		3ianor 1112	. . 3 ⊢ (¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
7	5, 6	bitri 276	. 2 ⊢ (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
8		df-ne 2935	. 2 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩)
9		df-ne 2935	. . 3 ⊢ (𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷)
10		df-ne 2935	. . 3 ⊢ (𝐵 ≠ 𝐸 ↔ ¬ 𝐵 = 𝐸)
11		df-ne 2935	. . 3 ⊢ (𝐶 ≠ 𝐹 ↔ ¬ 𝐶 = 𝐹)
12	9, 10, 11	3orbi123i 1162	. 2 ⊢ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
13	7, 8, 12	3bitr4i 304	1 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∨ w3o 1091   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431  ⟨cotp 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-ot 4564

This theorem is referenced by:  xpord3lem  8089  xpord3pred  8092  xpord3inddlem  8094