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Theorem sotricim 4114
Description: One direction of sotritric 4115 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 4108 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 463 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
323adant3 959 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  -.  B R B )
4 breq2 3815 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
54biimprcd 158 . . . . . 6  |-  ( B R C  ->  ( B  =  C  ->  B R B ) )
653ad2ant3 962 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  ( B  =  C  ->  B R B ) )
73, 6mtod 622 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  -.  B  =  C )
873expia 1141 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  B  =  C )
)
9 so2nr 4112 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
10 imnan 657 . . . 4  |-  ( ( B R C  ->  -.  C R B )  <->  -.  ( B R C  /\  C R B ) )
119, 10sylibr 132 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  C R B ) )
128, 11jcad 301 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  ( -.  B  =  C  /\  -.  C R B ) ) )
13 ioran 702 . 2  |-  ( -.  ( B  =  C  \/  C R B )  <->  ( -.  B  =  C  /\  -.  C R B ) )
1412, 13syl6ibr 160 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 102    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3811    Or wor 4086
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 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-po 4087  df-iso 4088
This theorem is referenced by:  sotritric  4115
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