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Theorem sotrieq 4827
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 4821 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 716 . . . . . 6  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 pm1.2 513 . . . . . 6  |-  ( ( B R B  \/  B R B )  ->  B R B )
42, 3nsyl 121 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R B  \/  B R B ) )
5 breq2 4451 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  B R C ) )
6 breq1 4450 . . . . . . 7  |-  ( B  =  C  ->  ( B R B  <->  C R B ) )
75, 6orbi12d 709 . . . . . 6  |-  ( B  =  C  ->  (
( B R B  \/  B R B )  <->  ( B R C  \/  C R B ) ) )
87notbid 294 . . . . 5  |-  ( B  =  C  ->  ( -.  ( B R B  \/  B R B )  <->  -.  ( B R C  \/  C R B ) ) )
94, 8syl5ibcom 220 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  ->  -.  ( B R C  \/  C R B ) ) )
109con2d 115 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  \/  C R B )  ->  -.  B  =  C ) )
11 solin 4823 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
12 3orass 976 . . . . 5  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( B R C  \/  ( B  =  C  \/  C R B ) ) )
13 or12 523 . . . . 5  |-  ( ( B R C  \/  ( B  =  C  \/  C R B ) )  <->  ( B  =  C  \/  ( B R C  \/  C R B ) ) )
14 df-or 370 . . . . 5  |-  ( ( B  =  C  \/  ( B R C  \/  C R B ) )  <-> 
( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1512, 13, 143bitri 271 . . . 4  |-  ( ( B R C  \/  B  =  C  \/  C R B )  <->  ( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1611, 15sylib 196 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( -.  B  =  C  ->  ( B R C  \/  C R B ) ) )
1710, 16impbid 191 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  \/  C R B )  <->  -.  B  =  C ) )
1817con2bid 329 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 184    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   class class class wbr 4447    Or wor 4799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-po 4800  df-so 4801
This theorem is referenced by:  sotrieq2  4828  sossfld  5454  soisores  6211  soisoi  6212  weniso  6238  wemapsolem  7975  distrlem4pr  9404  addcanpr  9424  sqgt0sr  9483  lttri2  9667  xrlttri2  11348  xrltne  11366  soseq  28939
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