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Theorem sotricim 4161
Description: One direction of sotritric 4162 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 4155 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 464 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
323adant3 964 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  -.  B R B )
4 breq2 3857 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
54biimprcd 159 . . . . . 6  |-  ( B R C  ->  ( B  =  C  ->  B R B ) )
653ad2ant3 967 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  ( B  =  C  ->  B R B ) )
73, 6mtod 625 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  -.  B  =  C )
873expia 1146 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  B  =  C )
)
9 so2nr 4159 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
10 imnan 660 . . . 4  |-  ( ( B R C  ->  -.  C R B )  <->  -.  ( B R C  /\  C R B ) )
119, 10sylibr 133 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  C R B ) )
128, 11jcad 302 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  ( -.  B  =  C  /\  -.  C R B ) ) )
13 ioran 705 . 2  |-  ( -.  ( B  =  C  \/  C R B )  <->  ( -.  B  =  C  /\  -.  C R B ) )
1412, 13syl6ibr 161 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 103    \/ wo 665    /\ w3a 925    = wceq 1290    e. wcel 1439   class class class wbr 3853    Or wor 4133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-un 3006  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-po 4134  df-iso 4135
This theorem is referenced by:  sotritric  4162
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