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Theorem so2nr 4211
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 4203 . 2  |-  ( R  Or  A  ->  R  Po  A )
2 po2nr 4199 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
31, 2sylan 279 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 103    e. wcel 1463   class class class wbr 3897    Po wpo 4184    Or wor 4185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-po 4186  df-iso 4187
This theorem is referenced by:  sotricim  4213  cauappcvgprlemdisj  7423  cauappcvgprlemladdru  7428  cauappcvgprlemladdrl  7429  caucvgprlemnbj  7439  caucvgprprlemnbj  7465  suplocexprlemmu  7490  ltnsym2  7818
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