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Theorem sotritric 4052
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 4051 . . 3 ((𝑅 Or A (B A 𝐶 A)) → (B𝑅𝐶 → ¬ (B = 𝐶 𝐶𝑅B)))
31, 2mpan 400 . 2 ((B A 𝐶 A) → (B𝑅𝐶 → ¬ (B = 𝐶 𝐶𝑅B)))
4 sotritric.tri . . 3 ((B A 𝐶 A) → (B𝑅𝐶 B = 𝐶 𝐶𝑅B))
5 3orass 887 . . . 4 ((B𝑅𝐶 B = 𝐶 𝐶𝑅B) ↔ (B𝑅𝐶 (B = 𝐶 𝐶𝑅B)))
6 ax-1 5 . . . . 5 (B𝑅𝐶 → (¬ (B = 𝐶 𝐶𝑅B) → B𝑅𝐶))
7 pm2.24 551 . . . . 5 ((B = 𝐶 𝐶𝑅B) → (¬ (B = 𝐶 𝐶𝑅B) → B𝑅𝐶))
86, 7jaoi 635 . . . 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 628   w3o 883   = wceq 1242   wcel 1390   class class class wbr 3755   Or wor 4023
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024  df-iso 4025
This theorem is referenced by:  nqtric  6383
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