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Theorem sotrine 5584
Description: Trichotomy law for strict orderings. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
sotrine ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 ↔ (𝐵𝑅𝐶𝐶𝑅𝐵)))

Proof of Theorem sotrine
StepHypRef Expression
1 sotrieq 5575 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
21bicomd 222 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ 𝐵 = 𝐶))
32necon1abid 2983 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 ↔ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2944   class class class wbr 5106   Or wor 5545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-po 5546  df-so 5547
This theorem is referenced by:  nosepne  27031  nosepdm  27035  slttrine  27102
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