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Theorem po3nr 4769
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 4768 . . 3  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  D  e.  A
) )  ->  -.  ( B R D  /\  D R B ) )
213adantr2 1168 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R D  /\  D R B ) )
3 df-3an 987 . . 3  |-  ( ( B R C  /\  C R D  /\  D R B )  <->  ( ( B R C  /\  C R D )  /\  D R B ) )
4 potr 4767 . . . 4  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
54anim1d 568 . . 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 221 . 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 181 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    e. wcel 1887   class class class wbr 4402    Po wpo 4753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-po 4755
This theorem is referenced by:  so3nr  4780
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