Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sotrine Structured version   Visualization version   GIF version

Theorem sotrine 33453
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 5497 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
21bicomd 226 . 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 209  wa 399  wo 847   = wceq 1543  wcel 2110  wne 2940   class class class wbr 5053   Or wor 5467
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 2112  ax-9 2120  ax-ext 2708
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 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-po 5468  df-so 5469
This theorem is referenced by:  nosepne  33620  nosepdm  33624  slttrine  33691
  Copyright terms: Public domain W3C validator