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Theorem sotrine 5632
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 5623 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
21bicomd 223 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ 𝐵 = 𝐶))
32necon1abid 2979 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 ↔ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143   Or wor 5591
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-po 5592  df-so 5593
This theorem is referenced by:  nosepne  27725  nosepdm  27729  slttrine  27796
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