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Theorem sotritric 4321
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 4320 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
31, 2mpan 424 . 2 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
4 sotritric.tri . . 3 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
5 3orass 981 . . . 4 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
6 ax-1 6 . . . . 5 (𝐵𝑅𝐶 → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
7 pm2.24 621 . . . . 5 ((𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
86, 7jaoi 716 . . . 4 ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
95, 8sylbi 121 . . 3 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
104, 9syl 14 . 2 ((𝐵𝐴𝐶𝐴) → (¬ (𝐵 = 𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐶))
113, 10impbid 129 1 ((𝐵𝐴𝐶𝐴) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
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 4000   Or wor 4292
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 3597  df-pr 3598  df-op 3600  df-br 4001  df-po 4293  df-iso 4294
This theorem is referenced by:  nqtric  7386
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