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Theorem sotritrieq 4325
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 4317 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
31, 2mpan 424 . . . . . 6 (𝐵𝐴 → ¬ 𝐵𝑅𝐵)
4 breq2 4007 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
54notbid 667 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
63, 5syl5ibcom 155 . . . . 5 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
7 breq1 4006 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐶𝑅𝐵))
87notbid 667 . . . . . 6 (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
93, 8syl5ibcom 155 . . . . 5 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ 𝐶𝑅𝐵))
106, 9jcad 307 . . . 4 (𝐵𝐴 → (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
11 ioran 752 . . . 4 (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
1210, 11imbitrrdi 162 . . 3 (𝐵𝐴 → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1312adantr 276 . 2 ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
14 sotritric.tri . . 3 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
15 3orrot 984 . . . . . . 7 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶𝐶𝑅𝐵𝐵𝑅𝐶))
16 3orcomb 987 . . . . . . 7 ((𝐵 = 𝐶𝐶𝑅𝐵𝐵𝑅𝐶) ↔ (𝐵 = 𝐶𝐵𝑅𝐶𝐶𝑅𝐵))
17 3orass 981 . . . . . . 7 ((𝐵 = 𝐶𝐵𝑅𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1815, 16, 173bitri 206 . . . . . 6 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1918biimpi 120 . . . . 5 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
2019orcomd 729 . . . 4 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → ((𝐵𝑅𝐶𝐶𝑅𝐵) ∨ 𝐵 = 𝐶))
2120ord 724 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵 = 𝐶))
2214, 21syl 14 . 2 ((𝐵𝐴𝐶𝐴) → (¬ (𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵 = 𝐶))
2313, 22impbid 129 1 ((𝐵𝐴𝐶𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  w3o 977   = wceq 1353  wcel 2148   class class class wbr 4003   Or wor 4295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-po 4296  df-iso 4297
This theorem is referenced by:  distrlem4prl  7582  distrlem4pru  7583
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