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Theorem sotrieq2 4466
Description: Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.)
Assertion
Ref Expression
sotrieq2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  ( -.  B R C  /\  -.  C R B ) ) )

Proof of Theorem sotrieq2
StepHypRef Expression
1 sotrieq 4465 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  -.  ( B R C  \/  C R B ) ) )
2 ioran 477 . 2  |-  ( -.  ( B R C  \/  C R B )  <->  ( -.  B R C  /\  -.  C R B ) )
31, 2syl6bb 253 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    = wceq 1649    e. wcel 1717   class class class wbr 4147    Or wor 4437
This theorem is referenced by:  fisupg  7285  supmo  7384  supmax  7397  lttri3  9085  xrlttri3  10662  slttrieq2  25346
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
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 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ral 2648  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-br 4148  df-po 4438  df-so 4439
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