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Theorem sotritric 4087
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  e.  A  /\  C  e.  A )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
Assertion
Ref Expression
sotritric  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B R C  <->  -.  ( B  =  C  \/  C R B ) ) )

Proof of Theorem sotritric
StepHypRef Expression
1 sotritric.or . . 3  |-  R  Or  A
2 sotricim 4086 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  ( B  =  C  \/  C R B ) ) )
31, 2mpan 415 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B R C  ->  -.  ( B  =  C  \/  C R B ) ) )
4 sotritric.tri . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
5 3orass 923 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B R C  \/  ( B  =  C  \/  C R B ) ) )
6 ax-1 5 . . . . 5  |-  ( B R C  ->  ( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
7 pm2.24 584 . . . . 5  |-  ( ( B  =  C  \/  C R B )  -> 
( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
86, 7jaoi 669 . . . 4  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  ->  ( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
95, 8sylbi 119 . . 3  |-  ( ( B R C  \/  B  =  C  \/  C R B )  -> 
( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
104, 9syl 14 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
113, 10impbid 127 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B R C  <->  -.  ( B  =  C  \/  C R B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    \/ w3o 919    = wceq 1285    e. wcel 1434   class class class wbr 3793    Or wor 4058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-po 4059  df-iso 4060
This theorem is referenced by:  nqtric  6651
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