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Theorem sotrieq 4787
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 4781 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 731 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 pm1.2 522 . . . . . 6  |-  ( ( B R B  \/  B R B )  ->  B R B )
42, 3nsyl 125 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R B  \/  B R B ) )
5 breq2 4399 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
6 breq1 4398 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  C R B ) )
75, 6orbi12d 724 . . . . . 6  |-  ( B  =  C  ->  (
( B R B  \/  B R B )  <->  ( B R C  \/  C R B ) ) )
87notbid 301 . . . . 5  |-  ( B  =  C  ->  ( -.  ( B R B  \/  B R B )  <->  -.  ( B R C  \/  C R B ) ) )
94, 8syl5ibcom 228 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  ->  -.  ( B R C  \/  C R B ) ) )
109con2d 119 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  \/  C R B )  ->  -.  B  =  C ) )
11 solin 4783 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
12 3orass 1010 . . . . 5  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B R C  \/  ( B  =  C  \/  C R B ) ) )
13 or12 532 . . . . 5  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  <->  ( B  =  C  \/  ( B R C  \/  C R B ) ) )
14 df-or 377 . . . . 5  |-  ( ( B  =  C  \/  ( B R C  \/  C R B ) )  <-> 
( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1512, 13, 143bitri 279 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1611, 15sylib 201 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1710, 16impbid 195 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  \/  C R B )  <->  -.  B  =  C ) )
1817con2bid 336 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  -.  ( B R C  \/  C R B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    \/ w3o 1006    = wceq 1452    e. wcel 1904   class class class wbr 4395    Or wor 4759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-po 4760  df-so 4761
This theorem is referenced by:  sotrieq2  4788  sossfld  5289  soisores  6236  soisoi  6237  weniso  6263  wemapsolem  8083  distrlem4pr  9469  addcanpr  9489  sqgt0sr  9548  lttri2  9734  xrlttri2  11464  xrltne  11483  soseq  30563
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