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Theorem sotric 4740
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 4735 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
2 breq2 4371 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
32notbid 292 . . . . . 6  |-  ( B  =  C  ->  ( -.  B R B  <->  -.  B R C ) )
41, 3syl5ibcom 220 . . . . 5  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( B  =  C  ->  -.  B R C ) )
54adantrr 714 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  ->  -.  B R C ) )
6 so2nr 4738 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
7 imnan 420 . . . . . 6  |-  ( ( B R C  ->  -.  C R B )  <->  -.  ( B R C  /\  C R B ) )
86, 7sylibr 212 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  ->  -.  C R B ) )
98con2d 115 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C R B  ->  -.  B R C ) )
105, 9jaod 378 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  =  C  \/  C R B )  ->  -.  B R C ) )
11 solin 4737 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
12 3orass 974 . . . . 5  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B R C  \/  ( B  =  C  \/  C R B ) ) )
13 df-or 368 . . . . 5  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  <->  ( -.  B R C  ->  ( B  =  C  \/  C R B ) ) )
1412, 13bitri 249 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( -.  B R C  ->  ( B  =  C  \/  C R B ) ) )
1511, 14sylib 196 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B R C  -> 
( B  =  C  \/  C R B ) ) )
1610, 15impbid 191 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  =  C  \/  C R B )  <->  -.  B R C ) )
1716con2bid 327 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  <->  -.  ( B  =  C  \/  C R B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    \/ w3o 970    = wceq 1399    e. wcel 1826   class class class wbr 4367    Or wor 4713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-po 4714  df-so 4715
This theorem is referenced by:  sotr2  4743  sotri2  5309  sotri3  5310  somin1  5313  somincom  5314  soisores  6124  soisoi  6125  fimaxg  7682  suplub2  7835  supgtoreq  7843  ordtypelem7  7864  fpwwe2  8932  indpi  9196  nqereu  9218  ltsonq  9258  prub  9283  ltapr  9334  suplem2pr  9342  ltsosr  9382  axpre-lttri  9453
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