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Theorem sotritrieq 4247
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 4239 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
31, 2mpan 420 . . . . . 6  |-  ( B  e.  A  ->  -.  B R B )
4 breq2 3933 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
54notbid 656 . . . . . 6  |-  ( B  =  C  ->  ( -.  B R B  <->  -.  B R C ) )
63, 5syl5ibcom 154 . . . . 5  |-  ( B  e.  A  ->  ( B  =  C  ->  -.  B R C ) )
7 breq1 3932 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  C R B ) )
87notbid 656 . . . . . 6  |-  ( B  =  C  ->  ( -.  B R B  <->  -.  C R B ) )
93, 8syl5ibcom 154 . . . . 5  |-  ( B  e.  A  ->  ( B  =  C  ->  -.  C R B ) )
106, 9jcad 305 . . . 4  |-  ( B  e.  A  ->  ( B  =  C  ->  ( -.  B R C  /\  -.  C R B ) ) )
11 ioran 741 . . . 4  |-  ( -.  ( B R C  \/  C R B )  <->  ( -.  B R C  /\  -.  C R B ) )
1210, 11syl6ibr 161 . . 3  |-  ( B  e.  A  ->  ( B  =  C  ->  -.  ( B R C  \/  C R B ) ) )
1312adantr 274 . 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 968 . . . . . . 7  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B  =  C  \/  C R B  \/  B R C ) )
16 3orcomb 971 . . . . . . 7  |-  ( ( B  =  C  \/  C R B  \/  B R C )  <->  ( B  =  C  \/  B R C  \/  C R B ) )
17 3orass 965 . . . . . . 7  |-  ( ( B  =  C  \/  B R C  \/  C R B )  <->  ( B  =  C  \/  ( B R C  \/  C R B ) ) )
1815, 16, 173bitri 205 . . . . . 6  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B  =  C  \/  ( B R C  \/  C R B ) ) )
1918biimpi 119 . . . . 5  |-  ( ( B R C  \/  B  =  C  \/  C R B )  -> 
( B  =  C  \/  ( B R C  \/  C R B ) ) )
2019orcomd 718 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  -> 
( ( B R C  \/  C R B )  \/  B  =  C ) )
2120ord 713 . . 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 128 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 103    <-> wb 104    \/ wo 697    \/ w3o 961    = wceq 1331    e. wcel 1480   class class class wbr 3929    Or wor 4217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-po 4218  df-iso 4219
This theorem is referenced by:  distrlem4prl  7399  distrlem4pru  7400
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