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Theorem sotric 5626
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 5621 . . . . . 6 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
2 breq2 5152 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
32notbid 318 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
41, 3syl5ibcom 245 . . . . 5 ((𝑅 Or 𝐴𝐵𝐴) → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
54adantrr 717 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
6 so2nr 5624 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
7 imnan 399 . . . . . 6 ((𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
86, 7sylibr 234 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ 𝐶𝑅𝐵))
98con2d 134 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝑅𝐵 → ¬ 𝐵𝑅𝐶))
105, 9jaod 859 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 = 𝐶𝐶𝑅𝐵) → ¬ 𝐵𝑅𝐶))
11 solin 5623 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
12 3orass 1089 . . . . 5 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
1311, 12sylib 218 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
1413ord 864 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐵𝑅𝐶 → (𝐵 = 𝐶𝐶𝑅𝐵)))
1510, 14impbid 212 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 = 𝐶𝐶𝑅𝐵) ↔ ¬ 𝐵𝑅𝐶))
1615con2bid 354 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3o 1085   = wceq 1537  wcel 2106   class class class wbr 5148   Or wor 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-po 5597  df-so 5598
This theorem is referenced by:  soasym  5629  sotr2  5630  sotr3  5637  sotri2  6152  sotri3  6153  somin1  6156  somincom  6157  soisores  7347  soisoi  7348  fimaxg  9321  suplub2  9499  supgtoreq  9508  fiming  9536  infsupprpr  9542  ordtypelem7  9562  fpwwe2  10681  indpi  10945  nqereu  10967  ltsonq  11007  prub  11032  ltapr  11083  suplem2pr  11091  ltsosr  11132  axpre-lttri  11203  noetasuplem4  27796  noetainflem4  27800  sleloe  27814  prproropf1olem4  47431
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