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Theorem sotritric 4058
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 4057 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
31, 2mpan 400 . 2 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
4 sotritric.tri . . 3 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
5 3orass 888 . . . 4 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
6 ax-1 5 . . . . 5 (𝐵𝑅𝐶 → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
7 pm2.24 551 . . . . 5 ((𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
86, 7jaoi 636 . . . 4 ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
95, 8sylbi 114 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
104, 9syl 14 . 2 ((𝐵𝐴𝐶𝐴) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
113, 10impbid 120 1 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
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 3761   Or wor 4029
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 3762  df-po 4030  df-iso 4031
This theorem is referenced by:  nqtric  6454
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