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Theorem sotric 4356
Description: A strict order relation satisfies strict trichotomy. (Contributed by NM, 19-Feb-1996.)
Assertion
Ref Expression
sotric  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  <->  -.  ( B  =  C  \/  C R B ) ) )

Proof of Theorem sotric
StepHypRef Expression
1 sonr 4351 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
2 breq2 4043 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
32notbid 285 . . . . . 6  |-  ( B  =  C  ->  ( -.  B R B  <->  -.  B R C ) )
41, 3syl5ibcom 211 . . . . 5  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( B  =  C  ->  -.  B R C ) )
54adantrr 697 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  ->  -.  B R C ) )
6 so2nr 4354 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
7 imnan 411 . . . . . 6  |-  ( ( B R C  ->  -.  C R B )  <->  -.  ( B R C  /\  C R B ) )
86, 7sylibr 203 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  C R B ) )
98con2d 107 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C R B  ->  -.  B R C ) )
105, 9jaod 369 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  =  C  \/  C R B )  ->  -.  B R C ) )
11 solin 4353 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
12 3orass 937 . . . . 5  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B R C  \/  ( B  =  C  \/  C R B ) ) )
13 df-or 359 . . . . 5  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  <->  ( -.  B R C  ->  ( B  =  C  \/  C R B ) ) )
1412, 13bitri 240 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( -.  B R C  ->  ( B  =  C  \/  C R B ) ) )
1511, 14sylib 188 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B R C  -> 
( B  =  C  \/  C R B ) ) )
1610, 15impbid 183 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  =  C  \/  C R B )  <->  -.  B R C ) )
1716con2bid 319 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    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696   class class class wbr 4039    Or wor 4329
This theorem is referenced by:  sotr2  4359  sotri2  5088  sotri3  5089  somin1  5095  somincom  5096  soisores  5840  soisoi  5841  fimaxg  7120  suplub2  7228  ordtypelem7  7255  fpwwe2  8281  indpi  8547  nqereu  8569  ltsonq  8609  prub  8634  ltapr  8685  suplem2pr  8693  ltsosr  8732  axpre-lttri  8803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-po 4330  df-so 4331
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