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Theorem sotritrieq 4119
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 4111 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
31, 2mpan 415 . . . . . 6 (𝐵𝐴 → ¬ 𝐵𝑅𝐵)
4 breq2 3818 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
54notbid 625 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
63, 5syl5ibcom 153 . . . . 5 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
7 breq1 3817 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐶𝑅𝐵))
87notbid 625 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
93, 8syl5ibcom 153 . . . . 5 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ 𝐶𝑅𝐵))
106, 9jcad 301 . . . 4 (𝐵𝐴 → (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
11 ioran 702 . . . 4 (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
1210, 11syl6ibr 160 . . 3 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1312adantr 270 . 2 ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
14 sotritric.tri . . 3 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
15 3orrot 928 . . . . . . 7 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶𝐶𝑅𝐵𝐵𝑅𝐶))
16 3orcomb 931 . . . . . . 7 ((𝐵 = 𝐶𝐶𝑅𝐵𝐵𝑅𝐶) ↔ (𝐵 = 𝐶𝐵𝑅𝐶𝐶𝑅𝐵))
17 3orass 925 . . . . . . 7 ((𝐵 = 𝐶𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1815, 16, 173bitri 204 . . . . . 6 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1918biimpi 118 . . . . 5 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
2019orcomd 681 . . . 4 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → ((𝐵𝑅𝐶𝐶𝑅𝐵) ∨ 𝐵 = 𝐶))
2120ord 676 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵 = 𝐶))
2214, 21syl 14 . 2 ((𝐵𝐴𝐶𝐴) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵 = 𝐶))
2313, 22impbid 127 1 ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3o 921   = wceq 1287  wcel 1436   class class class wbr 3814   Or wor 4089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-un 2990  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-po 4090  df-iso 4091
This theorem is referenced by:  distrlem4prl  7064  distrlem4pru  7065
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