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Theorem sotritric 4115
Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 28-Sep-2019.)
Hypotheses
Ref Expression
sotritric.or 𝑅 Or 𝐴
sotritric.tri ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
Assertion
Ref Expression
sotritric ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))

Proof of Theorem sotritric
StepHypRef Expression
1 sotritric.or . . 3 𝑅 Or 𝐴
2 sotricim 4114 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
31, 2mpan 415 . 2 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
4 sotritric.tri . . 3 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
5 3orass 923 . . . 4 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
6 ax-1 5 . . . . 5 (𝐵𝑅𝐶 → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
7 pm2.24 584 . . . . 5 ((𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
86, 7jaoi 669 . . . 4 ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
95, 8sylbi 119 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
104, 9syl 14 . 2 ((𝐵𝐴𝐶𝐴) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
113, 10impbid 127 1 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3o 919   = wceq 1285  wcel 1434   class class class wbr 3811   Or wor 4086
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-po 4087  df-iso 4088
This theorem is referenced by:  nqtric  6861
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