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Theorem so2nr 4339
Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)
Assertion
Ref Expression
so2nr  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )

Proof of Theorem so2nr
StepHypRef Expression
1 sopo 4331 . 2  |-  ( R  Or  A  ->  R  Po  A )
2 po2nr 4327 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
31, 2sylan 283 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    e. wcel 2160   class class class wbr 4018    Po wpo 4312    Or wor 4313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314  df-iso 4315
This theorem is referenced by:  sotricim  4341  cauappcvgprlemdisj  7680  cauappcvgprlemladdru  7685  cauappcvgprlemladdrl  7686  caucvgprlemnbj  7696  caucvgprprlemnbj  7722  suplocexprlemmu  7747  ltnsym2  8078
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