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Theorem sotrieq 4357
Description: Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
sotrieq  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  -.  ( B R C  \/  C R B ) ) )

Proof of Theorem sotrieq
StepHypRef Expression
1 sonr 4351 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 697 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 pm1.2 499 . . . . . 6  |-  ( ( B R B  \/  B R B )  ->  B R B )
42, 3nsyl 113 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R B  \/  B R B ) )
5 breq2 4043 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
6 breq1 4042 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  C R B ) )
75, 6orbi12d 690 . . . . . 6  |-  ( B  =  C  ->  (
( B R B  \/  B R B )  <->  ( B R C  \/  C R B ) ) )
87notbid 285 . . . . 5  |-  ( B  =  C  ->  ( -.  ( B R B  \/  B R B )  <->  -.  ( B R C  \/  C R B ) ) )
94, 8syl5ibcom 211 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  ->  -.  ( B R C  \/  C R B ) ) )
109con2d 107 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  \/  C R B )  ->  -.  B  =  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 or12 509 . . . . 5  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  <->  ( B  =  C  \/  ( B R C  \/  C R B ) ) )
14 df-or 359 . . . . 5  |-  ( ( B  =  C  \/  ( B R C  \/  C R B ) )  <-> 
( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1512, 13, 143bitri 262 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1611, 15sylib 188 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1710, 16impbid 183 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  \/  C R B )  <->  -.  B  =  C ) )
1817con2bid 319 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  -.  ( B R 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:  sotrieq2  4358  sossfld  5136  soisores  5840  soisoi  5841  weniso  5868  wemapso2lem  7281  distrlem4pr  8666  addcanpr  8686  sqgt0sr  8744  lttri2  8920  xrlttri2  10492  xrltne  10510  soseq  24325
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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