Theorem otthne 5468

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 5466	. . . 4 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))
5	4	notbii 323	. . 3 ⊢ (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))
6		3ianor 1124	. . 3 ⊢ (¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
7	5, 6	bitri 278	. 2 ⊢ (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
8		df-ne 2959	. 2 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩)
9		df-ne 2959	. . 3 ⊢ (𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷)
10		df-ne 2959	. . 3 ⊢ (𝐵 ≠ 𝐸 ↔ ¬ 𝐵 = 𝐸)
11		df-ne 2959	. . 3 ⊢ (𝐶 ≠ 𝐹 ↔ ¬ 𝐶 = 𝐹)
12	9, 10, 11	3orbi123i 1174	. 2 ⊢ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
13	7, 8, 12	3bitr4i 306	1 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ⟨cotp 4597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598

This theorem is referenced by:  xpord3lem  8141  xpord3pred  8144  xpord3inddlem  8146