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Theorem sotric 4816
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 4811 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
2 breq2 4441 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
32notbid 294 . . . . . 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 716 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  ->  -.  B R C ) )
6 so2nr 4814 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
7 imnan 422 . . . . . 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 380 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B  =  C  \/  C R B )  ->  -.  B R C ) )
11 solin 4813 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
12 3orass 977 . . . . 5  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B R C  \/  ( B  =  C  \/  C R B ) ) )
13 df-or 370 . . . . 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 329 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 368    /\ wa 369    \/ w3o 973    = wceq 1383    e. wcel 1804   class class class wbr 4437    Or wor 4789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-po 4790  df-so 4791
This theorem is referenced by:  sotr2  4819  sotri2  5386  sotri3  5387  somin1  5393  somincom  5394  soisores  6208  soisoi  6209  fimaxg  7769  suplub2  7923  supgtoreq  7931  ordtypelem7  7952  fpwwe2  9024  indpi  9288  nqereu  9310  ltsonq  9350  prub  9375  ltapr  9426  suplem2pr  9434  ltsosr  9474  axpre-lttri  9545
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