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Theorem sotrieq 5052
Description: Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
sotrieq ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))

Proof of Theorem sotrieq
StepHypRef Expression
1 sonr 5046 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 752 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
3 pm1.2 535 . . . . . 6 ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵)
42, 3nsyl 135 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐵𝐵𝑅𝐵))
5 breq2 4648 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
6 breq1 4647 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐶𝑅𝐵))
75, 6orbi12d 745 . . . . . 6 (𝐵 = 𝐶 → ((𝐵𝑅𝐵𝐵𝑅𝐵) ↔ (𝐵𝑅𝐶𝐶𝑅𝐵)))
87notbid 308 . . . . 5 (𝐵 = 𝐶 → (¬ (𝐵𝑅𝐵𝐵𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
94, 8syl5ibcom 235 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
109con2d 129 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → ¬ 𝐵 = 𝐶))
11 solin 5048 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
12 3orass 1039 . . . . . 6 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
1311, 12sylib 208 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
14 or12 545 . . . . 5 ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1513, 14sylib 208 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1615ord 392 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐵 = 𝐶 → (𝐵𝑅𝐶𝐶𝑅𝐵)))
1710, 16impbid 202 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) ↔ ¬ 𝐵 = 𝐶))
1817con2bid 344 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035   = wceq 1481  wcel 1988   class class class wbr 4644   Or wor 5024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-po 5025  df-so 5026
This theorem is referenced by:  sotrieq2  5053  sossfld  5568  soisores  6562  soisoi  6563  weniso  6589  wemapsolem  8440  distrlem4pr  9833  addcanpr  9853  sqgt0sr  9912  lttri2  10105  xrlttri2  11960  xrltne  11979  sotrine  31634  soseq  31725
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