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Theorem sotric 4529
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 4524 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
2 breq2 4216 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
32notbid 286 . . . . . 6  |-  ( B  =  C  ->  ( -.  B R B  <->  -.  B R C ) )
41, 3syl5ibcom 212 . . . . 5  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( B  =  C  ->  -.  B R C ) )
54adantrr 698 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  ->  -.  B R C ) )
6 so2nr 4527 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
7 imnan 412 . . . . . 6  |-  ( ( B R C  ->  -.  C R B )  <->  -.  ( B R C  /\  C R B ) )
86, 7sylibr 204 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  C R B ) )
98con2d 109 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C R B  ->  -.  B R C ) )
105, 9jaod 370 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  =  C  \/  C R B )  ->  -.  B R C ) )
11 solin 4526 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
12 3orass 939 . . . . 5  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B R C  \/  ( B  =  C  \/  C R B ) ) )
13 df-or 360 . . . . 5  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  <->  ( -.  B R C  ->  ( B  =  C  \/  C R B ) ) )
1412, 13bitri 241 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( -.  B R C  ->  ( B  =  C  \/  C R B ) ) )
1511, 14sylib 189 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B R C  -> 
( B  =  C  \/  C R B ) ) )
1610, 15impbid 184 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  =  C  \/  C R B )  <->  -.  B R C ) )
1716con2bid 320 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 177    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725   class class class wbr 4212    Or wor 4502
This theorem is referenced by:  sotr2  4532  sotri2  5263  sotri3  5264  somin1  5270  somincom  5271  soisores  6047  soisoi  6048  fimaxg  7354  suplub2  7466  ordtypelem7  7493  fpwwe2  8518  indpi  8784  nqereu  8806  ltsonq  8846  prub  8871  ltapr  8922  suplem2pr  8930  ltsosr  8969  axpre-lttri  9040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-po 4503  df-so 4504
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