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Theorem sotritric 4253
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 4252 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
31, 2mpan 421 . 2 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
4 sotritric.tri . . 3 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
5 3orass 966 . . . 4 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
6 ax-1 6 . . . . 5 (𝐵𝑅𝐶 → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
7 pm2.24 611 . . . . 5 ((𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
86, 7jaoi 706 . . . 4 ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
95, 8sylbi 120 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
104, 9syl 14 . 2 ((𝐵𝐴𝐶𝐴) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
113, 10impbid 128 1 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  w3o 962   = wceq 1332  wcel 1481   class class class wbr 3936   Or wor 4224
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-po 4225  df-iso 4226
This theorem is referenced by:  nqtric  7230
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