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Theorem sotricim 4306
Description: One direction of sotritric 4307 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 4300 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 476 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
323adant3 1012 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  -.  B R B )
4 breq2 3991 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
54biimprcd 159 . . . . . 6  |-  ( B R C  ->  ( B  =  C  ->  B R B ) )
653ad2ant3 1015 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  ( B  =  C  ->  B R B ) )
73, 6mtod 658 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
)  /\  B R C )  ->  -.  B  =  C )
873expia 1200 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  B  =  C )
)
9 so2nr 4304 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
10 imnan 685 . . . 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 305 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  ( -.  B  =  C  /\  -.  C R B ) ) )
13 ioran 747 . 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 703    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3987    Or wor 4278
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-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 3587  df-pr 3588  df-op 3590  df-br 3988  df-po 4279  df-iso 4280
This theorem is referenced by:  sotritric  4307
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