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Theorem sotritric 4285
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 4284 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  ( B  =  C  \/  C R B ) ) )
31, 2mpan 421 . 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 966 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B R C  \/  ( B  =  C  \/  C R B ) ) )
6 ax-1 6 . . . . 5  |-  ( B R C  ->  ( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
7 pm2.24 611 . . . . 5  |-  ( ( B  =  C  \/  C R B )  -> 
( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
86, 7jaoi 706 . . . 4  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  ->  ( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
95, 8sylbi 120 . . 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 128 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 103    <-> wb 104    \/ wo 698    \/ w3o 962    = wceq 1335    e. wcel 2128   class class class wbr 3966    Or wor 4256
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-po 4257  df-iso 4258
This theorem is referenced by:  nqtric  7320
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