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Theorem so3nr 4769
Description: A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)
Assertion
Ref Expression
so3nr  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )

Proof of Theorem so3nr
StepHypRef Expression
1 sopo 4761 . 2  |-  ( R  Or  A  ->  R  Po  A )
2 po3nr 4758 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
31, 2sylan 469 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    e. wcel 1842   class class class wbr 4395    Po wpo 4742    Or wor 4743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-po 4744  df-so 4745
This theorem is referenced by: (None)
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