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Theorem sotritrieq 4090
Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)
Hypotheses
Ref Expression
sotritric.or  |-  R  Or  A
sotritric.tri  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
Assertion
Ref Expression
sotritrieq  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B  =  C  <->  -.  ( B R C  \/  C R B ) ) )

Proof of Theorem sotritrieq
StepHypRef Expression
1 sotritric.or . . . . . . 7  |-  R  Or  A
2 sonr 4082 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
31, 2mpan 408 . . . . . 6  |-  ( B  e.  A  ->  -.  B R B )
4 breq2 3796 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
54notbid 602 . . . . . 6  |-  ( B  =  C  ->  ( -.  B R B  <->  -.  B R C ) )
63, 5syl5ibcom 148 . . . . 5  |-  ( B  e.  A  ->  ( B  =  C  ->  -.  B R C ) )
7 breq1 3795 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  C R B ) )
87notbid 602 . . . . . 6  |-  ( B  =  C  ->  ( -.  B R B  <->  -.  C R B ) )
93, 8syl5ibcom 148 . . . . 5  |-  ( B  e.  A  ->  ( B  =  C  ->  -.  C R B ) )
106, 9jcad 295 . . . 4  |-  ( B  e.  A  ->  ( B  =  C  ->  ( -.  B R C  /\  -.  C R B ) ) )
11 ioran 679 . . . 4  |-  ( -.  ( B R C  \/  C R B )  <->  ( -.  B R C  /\  -.  C R B ) )
1210, 11syl6ibr 155 . . 3  |-  ( B  e.  A  ->  ( B  =  C  ->  -.  ( B R C  \/  C R B ) ) )
1312adantr 265 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B  =  C  ->  -.  ( B R C  \/  C R B ) ) )
14 sotritric.tri . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
15 3orrot 902 . . . . . . 7  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B  =  C  \/  C R B  \/  B R C ) )
16 3orcomb 905 . . . . . . 7  |-  ( ( B  =  C  \/  C R B  \/  B R C )  <->  ( B  =  C  \/  B R C  \/  C R B ) )
17 3orass 899 . . . . . . 7  |-  ( ( B  =  C  \/  B R C  \/  C R B )  <->  ( B  =  C  \/  ( B R C  \/  C R B ) ) )
1815, 16, 173bitri 199 . . . . . 6  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B  =  C  \/  ( B R C  \/  C R B ) ) )
1918biimpi 117 . . . . 5  |-  ( ( B R C  \/  B  =  C  \/  C R B )  -> 
( B  =  C  \/  ( B R C  \/  C R B ) ) )
2019orcomd 658 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  -> 
( ( B R C  \/  C R B )  \/  B  =  C ) )
2120ord 653 . . 3  |-  ( ( B R C  \/  B  =  C  \/  C R B )  -> 
( -.  ( B R C  \/  C R B )  ->  B  =  C ) )
2214, 21syl 14 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( -.  ( B R C  \/  C R B )  ->  B  =  C ) )
2313, 22impbid 124 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B  =  C  <->  -.  ( B R C  \/  C R B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    \/ w3o 895    = wceq 1259    e. wcel 1409   class class class wbr 3792    Or wor 4060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-po 4061  df-iso 4062
This theorem is referenced by:  distrlem4prl  6740  distrlem4pru  6741
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