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Theorem sotric 4985
 Description: A strict order relation satisfies strict trichotomy. (Contributed by NM, 19-Feb-1996.)
Assertion
Ref Expression
sotric ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))

Proof of Theorem sotric
StepHypRef Expression
1 sonr 4980 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
2 breq2 4587 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
32notbid 307 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
41, 3syl5ibcom 234 . . . . 5 ((𝑅 Or 𝐴𝐵𝐴) → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
54adantrr 749 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
6 so2nr 4983 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
7 imnan 437 . . . . . 6 ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
86, 7sylibr 223 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
98con2d 128 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝑅𝐵 → ¬ 𝐵𝑅𝐶))
105, 9jaod 394 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 = 𝐶𝐶𝑅𝐵) → ¬ 𝐵𝑅𝐶))
11 solin 4982 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
12 3orass 1034 . . . . 5 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
1311, 12sylib 207 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
1413ord 391 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐵𝑅𝐶 → (𝐵 = 𝐶𝐶𝑅𝐵)))
1510, 14impbid 201 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 = 𝐶𝐶𝑅𝐵) ↔ ¬ 𝐵𝑅𝐶))
1615con2bid 343 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   class class class wbr 4583   Or wor 4958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-po 4959  df-so 4960 This theorem is referenced by:  sotr2  4988  sotri2  5444  sotri3  5445  somin1  5448  somincom  5449  soisores  6477  soisoi  6478  fimaxg  8092  suplub2  8250  supgtoreq  8259  fiming  8287  ordtypelem7  8312  fpwwe2  9344  indpi  9608  nqereu  9630  ltsonq  9670  prub  9695  ltapr  9746  suplem2pr  9754  ltsosr  9794  axpre-lttri  9865
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