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Theorem sotrieq 4530
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 4524 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 698 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 pm1.2 500 . . . . . 6  |-  ( ( B R B  \/  B R B )  ->  B R B )
42, 3nsyl 115 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R B  \/  B R B ) )
5 breq2 4216 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
6 breq1 4215 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  C R B ) )
75, 6orbi12d 691 . . . . . 6  |-  ( B  =  C  ->  (
( B R B  \/  B R B )  <->  ( B R C  \/  C R B ) ) )
87notbid 286 . . . . 5  |-  ( B  =  C  ->  ( -.  ( B R B  \/  B R B )  <->  -.  ( B R C  \/  C R B ) ) )
94, 8syl5ibcom 212 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  ->  -.  ( B R C  \/  C R B ) ) )
109con2d 109 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  \/  C R B )  ->  -.  B  =  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 or12 510 . . . . 5  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  <->  ( B  =  C  \/  ( B R C  \/  C R B ) ) )
14 df-or 360 . . . . 5  |-  ( ( B  =  C  \/  ( B R C  \/  C R B ) )  <-> 
( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1512, 13, 143bitri 263 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1611, 15sylib 189 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1710, 16impbid 184 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  \/  C R B )  <->  -.  B  =  C ) )
1817con2bid 320 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 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:  sotrieq2  4531  sossfld  5317  soisores  6047  soisoi  6048  weniso  6075  wemapso2lem  7519  distrlem4pr  8903  addcanpr  8923  sqgt0sr  8981  lttri2  9157  xrlttri2  10735  xrltne  10753  soseq  25529
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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