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Theorem sotritrieq 4058
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 4050 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
31, 2mpan 400 . . . . . 6 (𝐵𝐴 → ¬ 𝐵𝑅𝐵)
4 breq2 3764 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
54notbid 592 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
63, 5syl5ibcom 144 . . . . 5 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
7 breq1 3763 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐶𝑅𝐵))
87notbid 592 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
93, 8syl5ibcom 144 . . . . 5 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ 𝐶𝑅𝐵))
106, 9jcad 291 . . . 4 (𝐵𝐴 → (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
11 ioran 669 . . . 4 (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
1210, 11syl6ibr 151 . . 3 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1312adantr 261 . 2 ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
14 sotritric.tri . . 3 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
15 3orrot 891 . . . . . . 7 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶𝐶𝑅𝐵𝐵𝑅𝐶))
16 3orcomb 894 . . . . . . 7 ((𝐵 = 𝐶𝐶𝑅𝐵𝐵𝑅𝐶) ↔ (𝐵 = 𝐶𝐵𝑅𝐶𝐶𝑅𝐵))
17 3orass 888 . . . . . . 7 ((𝐵 = 𝐶𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1815, 16, 173bitri 195 . . . . . 6 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1918biimpi 113 . . . . 5 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
2019orcomd 648 . . . 4 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → ((𝐵𝑅𝐶𝐶𝑅𝐵) ∨ 𝐵 = 𝐶))
2120ord 643 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵 = 𝐶))
2214, 21syl 14 . 2 ((𝐵𝐴𝐶𝐴) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵 = 𝐶))
2313, 22impbid 120 1 ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wo 629  w3o 884   = wceq 1243  wcel 1393   class class class wbr 3760   Or wor 4028
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-un 2919  df-sn 3378  df-pr 3379  df-op 3381  df-br 3761  df-po 4029  df-iso 4030
This theorem is referenced by:  distrlem4prl  6625  distrlem4pru  6626
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