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Theorem sotritrieq 4247
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 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
Assertion
Ref Expression
sotritrieq ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))

Proof of Theorem sotritrieq
StepHypRef Expression
1 sotritric.or . . . . . . 7 𝑅 Or 𝐴
2 sonr 4239 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
31, 2mpan 420 . . . . . 6 (𝐵𝐴 → ¬ 𝐵𝑅𝐵)
4 breq2 3933 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
54notbid 656 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
63, 5syl5ibcom 154 . . . . 5 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
7 breq1 3932 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐶𝑅𝐵))
87notbid 656 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
93, 8syl5ibcom 154 . . . . 5 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ 𝐶𝑅𝐵))
106, 9jcad 305 . . . 4 (𝐵𝐴 → (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
11 ioran 741 . . . 4 (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
1210, 11syl6ibr 161 . . 3 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1312adantr 274 . 2 ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
14 sotritric.tri . . 3 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
15 3orrot 968 . . . . . . 7 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶𝐶𝑅𝐵𝐵𝑅𝐶))
16 3orcomb 971 . . . . . . 7 ((𝐵 = 𝐶𝐶𝑅𝐵𝐵𝑅𝐶) ↔ (𝐵 = 𝐶𝐵𝑅𝐶𝐶𝑅𝐵))
17 3orass 965 . . . . . . 7 ((𝐵 = 𝐶𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1815, 16, 173bitri 205 . . . . . 6 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1918biimpi 119 . . . . 5 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
2019orcomd 718 . . . 4 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → ((𝐵𝑅𝐶𝐶𝑅𝐵) ∨ 𝐵 = 𝐶))
2120ord 713 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵 = 𝐶))
2214, 21syl 14 . 2 ((𝐵𝐴𝐶𝐴) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵 = 𝐶))
2313, 22impbid 128 1 ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  w3o 961   = wceq 1331  wcel 1480   class class class wbr 3929   Or wor 4217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-po 4218  df-iso 4219
This theorem is referenced by:  distrlem4prl  7399  distrlem4pru  7400
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