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Theorem sotritric 4035
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 ((B A 𝐶 A) → (B𝑅𝐶 B = 𝐶 𝐶𝑅B))
Assertion
Ref Expression
sotritric ((B A 𝐶 A) → (B𝑅𝐶 ↔ ¬ (B = 𝐶 𝐶𝑅B)))

Proof of Theorem sotritric
StepHypRef Expression
1 sotritric.or . . 3 𝑅 Or A
2 sotricim 4034 . . 3 ((𝑅 Or A (B A 𝐶 A)) → (B𝑅𝐶 → ¬ (B = 𝐶 𝐶𝑅B)))
31, 2mpan 402 . 2 ((B A 𝐶 A) → (B𝑅𝐶 → ¬ (B = 𝐶 𝐶𝑅B)))
4 sotritric.tri . . 3 ((B A 𝐶 A) → (B𝑅𝐶 B = 𝐶 𝐶𝑅B))
5 3orass 876 . . . 4 ((B𝑅𝐶 B = 𝐶 𝐶𝑅B) ↔ (B𝑅𝐶 (B = 𝐶 𝐶𝑅B)))
6 ax-1 5 . . . . 5 (B𝑅𝐶 → (¬ (B = 𝐶 𝐶𝑅B) → B𝑅𝐶))
7 pm2.24 539 . . . . 5 ((B = 𝐶 𝐶𝑅B) → (¬ (B = 𝐶 𝐶𝑅B) → B𝑅𝐶))
86, 7jaoi 623 . . . 4 ((B𝑅𝐶 (B = 𝐶 𝐶𝑅B)) → (¬ (B = 𝐶 𝐶𝑅B) → B𝑅𝐶))
95, 8sylbi 114 . . 3 ((B𝑅𝐶 B = 𝐶 𝐶𝑅B) → (¬ (B = 𝐶 𝐶𝑅B) → B𝑅𝐶))
104, 9syl 14 . 2 ((B A 𝐶 A) → (¬ (B = 𝐶 𝐶𝑅B) → B𝑅𝐶))
113, 10impbid 120 1 ((B A 𝐶 A) → (B𝑅𝐶 ↔ ¬ (B = 𝐶 𝐶𝑅B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616   w3o 872   = wceq 1228   wcel 1374   class class class wbr 3738   Or wor 4006
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-po 4007  df-iso 4008
This theorem is referenced by:  nqtric  6258
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