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Theorem sotritrieq 4052
 Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)
Hypotheses
Ref Expression
sotritric.or 𝑅 Or A
sotritric.tri ((B A 𝐶 A) → (B𝑅𝐶 B = 𝐶 𝐶𝑅B))
Assertion
Ref Expression
sotritrieq ((B A 𝐶 A) → (B = 𝐶 ↔ ¬ (B𝑅𝐶 𝐶𝑅B)))

Proof of Theorem sotritrieq
StepHypRef Expression
1 sotritric.or . . . . . . 7 𝑅 Or A
2 sonr 4044 . . . . . . 7 ((𝑅 Or A B A) → ¬ B𝑅B)
31, 2mpan 400 . . . . . 6 (B A → ¬ B𝑅B)
4 breq2 3758 . . . . . . 7 (B = 𝐶 → (B𝑅BB𝑅𝐶))
54notbid 591 . . . . . 6 (B = 𝐶 → (¬ B𝑅B ↔ ¬ B𝑅𝐶))
63, 5syl5ibcom 144 . . . . 5 (B A → (B = 𝐶 → ¬ B𝑅𝐶))
7 breq1 3757 . . . . . . 7 (B = 𝐶 → (B𝑅B𝐶𝑅B))
87notbid 591 . . . . . 6 (B = 𝐶 → (¬ B𝑅B ↔ ¬ 𝐶𝑅B))
93, 8syl5ibcom 144 . . . . 5 (B A → (B = 𝐶 → ¬ 𝐶𝑅B))
106, 9jcad 291 . . . 4 (B A → (B = 𝐶 → (¬ B𝑅𝐶 ¬ 𝐶𝑅B)))
11 ioran 668 . . . 4 (¬ (B𝑅𝐶 𝐶𝑅B) ↔ (¬ B𝑅𝐶 ¬ 𝐶𝑅B))
1210, 11syl6ibr 151 . . 3 (B A → (B = 𝐶 → ¬ (B𝑅𝐶 𝐶𝑅B)))
1312adantr 261 . 2 ((B A 𝐶 A) → (B = 𝐶 → ¬ (B𝑅𝐶 𝐶𝑅B)))
14 sotritric.tri . . 3 ((B A 𝐶 A) → (B𝑅𝐶 B = 𝐶 𝐶𝑅B))
15 3orrot 890 . . . . . . 7 ((B𝑅𝐶 B = 𝐶 𝐶𝑅B) ↔ (B = 𝐶 𝐶𝑅B B𝑅𝐶))
16 3orcomb 893 . . . . . . 7 ((B = 𝐶 𝐶𝑅B B𝑅𝐶) ↔ (B = 𝐶 B𝑅𝐶 𝐶𝑅B))
17 3orass 887 . . . . . . 7 ((B = 𝐶 B𝑅𝐶 𝐶𝑅B) ↔ (B = 𝐶 (B𝑅𝐶 𝐶𝑅B)))
1815, 16, 173bitri 195 . . . . . 6 ((B𝑅𝐶 B = 𝐶 𝐶𝑅B) ↔ (B = 𝐶 (B𝑅𝐶 𝐶𝑅B)))
1918biimpi 113 . . . . 5 ((B𝑅𝐶 B = 𝐶 𝐶𝑅B) → (B = 𝐶 (B𝑅𝐶 𝐶𝑅B)))
2019orcomd 647 . . . 4 ((B𝑅𝐶 B = 𝐶 𝐶𝑅B) → ((B𝑅𝐶 𝐶𝑅B) B = 𝐶))
2120ord 642 . . 3 ((B𝑅𝐶 B = 𝐶 𝐶𝑅B) → (¬ (B𝑅𝐶 𝐶𝑅B) → B = 𝐶))
2214, 21syl 14 . 2 ((B A 𝐶 A) → (¬ (B𝑅𝐶 𝐶𝑅B) → B = 𝐶))
2313, 22impbid 120 1 ((B A 𝐶 A) → (B = 𝐶 ↔ ¬ (B𝑅𝐶 𝐶𝑅B)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   ∨ w3o 883   = wceq 1242   ∈ wcel 1390   class class class wbr 3754   Or wor 4022 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-po 4023  df-iso 4024 This theorem is referenced by:  distrlem4prl  6550  distrlem4pru  6551
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