Theorem otthne 5466

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 5464	. . . 4 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))
5	4	notbii 320	. . 3 ⊢ (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))
6		3ianor 1106	. . 3 ⊢ (¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
7	5, 6	bitri 275	. 2 ⊢ (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
8		df-ne 2934	. 2 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩)
9		df-ne 2934	. . 3 ⊢ (𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷)
10		df-ne 2934	. . 3 ⊢ (𝐵 ≠ 𝐸 ↔ ¬ 𝐵 = 𝐸)
11		df-ne 2934	. . 3 ⊢ (𝐶 ≠ 𝐹 ↔ ¬ 𝐶 = 𝐹)
12	9, 10, 11	3orbi123i 1156	. 2 ⊢ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
13	7, 8, 12	3bitr4i 303	1 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  Vcvv 3464  ⟨cotp 4614

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-ot 4615

This theorem is referenced by:  xpord3lem  8153  xpord3pred  8156  xpord3inddlem  8158