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Theorem po3nr 4328
Description: A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
po3nr  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )

Proof of Theorem po3nr
StepHypRef Expression
1 po2nr 4327 . . 3  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  D  e.  A
) )  ->  -.  ( B R D  /\  D R B ) )
213adantr2 1115 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R D  /\  D R B ) )
3 df-3an 936 . . 3  |-  ( ( B R C  /\  C R D  /\  D R B )  <->  ( ( B R C  /\  C R D )  /\  D R B ) )
4 potr 4326 . . . 4  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
54anim1d 547 . . 3  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( ( B R C  /\  C R D )  /\  D R B )  ->  ( B R D  /\  D R B ) ) )
63, 5syl5bi 208 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D  /\  D R B )  ->  ( B R D  /\  D R B ) ) )
72, 6mtod 168 1  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684   class class class wbr 4023    Po wpo 4312
This theorem is referenced by:  so3nr  4339
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-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
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