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Theorem sotritrieq 4053
Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)
Hypotheses
Ref Expression
sotritric.or  R  Or
sotritric.tri  C  R C  C  C R
Assertion
Ref Expression
sotritrieq  C  C  R C  C R

Proof of Theorem sotritrieq
StepHypRef Expression
1 sotritric.or . . . . . . 7  R  Or
2 sonr 4045 . . . . . . 7  R  Or  R
31, 2mpan 400 . . . . . 6  R
4 breq2 3759 . . . . . . 7  C  R  R C
54notbid 591 . . . . . 6  C  R  R C
63, 5syl5ibcom 144 . . . . 5  C  R C
7 breq1 3758 . . . . . . 7  C  R  C R
87notbid 591 . . . . . 6  C  R  C R
93, 8syl5ibcom 144 . . . . 5  C  C R
106, 9jcad 291 . . . 4  C  R C  C R
11 ioran 668 . . . 4  R C  C R  R C  C R
1210, 11syl6ibr 151 . . 3  C  R C  C R
1312adantr 261 . 2  C  C  R C  C R
14 sotritric.tri . . 3  C  R C  C  C R
15 3orrot 890 . . . . . . 7  R C  C  C R  C  C R  R C
16 3orcomb 893 . . . . . . 7  C  C R  R C  C  R C  C R
17 3orass 887 . . . . . . 7  C  R C  C R  C  R C  C R
1815, 16, 173bitri 195 . . . . . 6  R C  C  C R  C  R C  C R
1918biimpi 113 . . . . 5  R C  C  C R  C  R C  C R
2019orcomd 647 . . . 4  R C  C  C R  R C  C R  C
2120ord 642 . . 3  R C  C  C R  R C  C R  C
2214, 21syl 14 . 2  C  R C  C R  C
2313, 22impbid 120 1  C  C  R C  C R
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-bndl 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:  distrlem4prl  6560  distrlem4pru  6561
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