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Theorem sotricim 4060
Description: One direction of sotritric 4061 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)
Assertion
Ref Expression
sotricim  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  ( B  =  C  \/  C R B ) ) )

Proof of Theorem sotricim
StepHypRef Expression
1 sonr 4054 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 448 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
323adant3 924 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  -.  B R B )
4 breq2 3768 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
54biimprcd 149 . . . . . 6  |-  ( B R C  ->  ( B  =  C  ->  B R B ) )
653ad2ant3 927 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  ( B  =  C  ->  B R B ) )
73, 6mtod 589 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  -.  B  =  C )
873expia 1106 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  B  =  C )
)
9 so2nr 4058 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
10 imnan 624 . . . 4  |-  ( ( B R C  ->  -.  C R B )  <->  -.  ( B R C  /\  C R B ) )
119, 10sylibr 137 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  C R B ) )
128, 11jcad 291 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  ( -.  B  =  C  /\  -.  C R B ) ) )
13 ioran 669 . 2  |-  ( -.  ( B  =  C  \/  C R B )  <->  ( -.  B  =  C  /\  -.  C R B ) )
1412, 13syl6ibr 151 1  |-  ( ( R  Or  A  /\  ( 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 97    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764    Or wor 4032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-iso 4034
This theorem is referenced by:  sotritric  4061
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