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Theorem sotritric 4309
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 4308 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  ( B  =  C  \/  C R B ) ) )
31, 2mpan 422 . 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 976 . . . 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 pm2.24 616 . . . . 5  |-  ( ( B  =  C  \/  C R B )  -> 
( -.  ( B  =  C  \/  C R B )  ->  B R C ) )
86, 7jaoi 711 . . . 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 703    \/ w3o 972    = wceq 1348    e. wcel 2141   class class class wbr 3989    Or wor 4280
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-po 4281  df-iso 4282
This theorem is referenced by:  nqtric  7361
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