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Theorem sotrieq 4490
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 4484 . . . . . . 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 4176 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
6 breq1 4175 . . . . . . 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 4486 . . . 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 1649    e. wcel 1721   class class class wbr 4172    Or wor 4462
This theorem is referenced by:  sotrieq2  4491  sossfld  5276  soisores  6006  soisoi  6007  weniso  6034  wemapso2lem  7475  distrlem4pr  8859  addcanpr  8879  sqgt0sr  8937  lttri2  9113  xrlttri2  10691  xrltne  10709  soseq  25468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-po 4463  df-so 4464
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