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Theorem sotrieq2 4342
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 4341 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  -.  ( B R C  \/  C R B ) ) )
2 ioran 476 . 2  |-  ( -.  ( B R C  \/  C R B )  <->  ( -.  B R C  /\  -.  C R B ) )
31, 2syl6bb 252 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023    Or wor 4313
This theorem is referenced by:  fisupg  7105  supmo  7203  supmax  7216  lttri3  8905  xrlttri3  10477  slttrieq2  24328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-po 4314  df-so 4315
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