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Theorem otthne 33428
Description: Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 21-Aug-2024.)
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⟩ ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))

Proof of Theorem otthne
StepHypRef Expression
1 otthne.1 . . . 4 𝐴 ∈ V
2 otthne.2 . . . 4 𝐵 ∈ V
31, 2opthne 5382 . . 3 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐷, 𝐸⟩ ↔ (𝐴𝐷𝐵𝐸))
43orbi1i 914 . 2 ((⟨𝐴, 𝐵⟩ ≠ ⟨𝐷, 𝐸⟩ ∨ 𝐶𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∨ 𝐶𝐹))
5 opex 5364 . . 3 𝐴, 𝐵⟩ ∈ V
6 otthne.3 . . 3 𝐶 ∈ V
75, 6opthne 5382 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ≠ ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐷, 𝐸⟩ ∨ 𝐶𝐹))
8 df-3or 1090 . 2 ((𝐴𝐷𝐵𝐸𝐶𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∨ 𝐶𝐹))
94, 7, 83bitr4i 306 1 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ≠ ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wo 847  w3o 1088  wcel 2112  wne 2943  Vcvv 3423  cop 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564
This theorem is referenced by:  xpord3lem  33565  xpord3pred  33568  xpord3ind  33570
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