MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sotrieq2 Structured version   Unicode version

Theorem sotrieq2 4531
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 4530 . 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 1652    e. wcel 1725   class class class wbr 4212    Or wor 4502
This theorem is referenced by:  fisupg  7355  supmo  7457  supmax  7470  lttri3  9158  xrlttri3  10736  slttrieq2  25629
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
  Copyright terms: Public domain W3C validator