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Theorem po3nr 4202
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 4201 . . 3  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  D  e.  A
) )  ->  -.  ( B R D  /\  D R B ) )
213adantr2 1126 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R D  /\  D R B ) )
3 df-3an 949 . . 3  |-  ( ( B R C  /\  C R D  /\  D R B )  <->  ( ( B R C  /\  C R D )  /\  D R B ) )
4 potr 4200 . . . 4  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
54anim1d 334 . . 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 151 . 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 637 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 103    /\ w3a 947    e. wcel 1465   class class class wbr 3899    Po wpo 4186
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-po 4188
This theorem is referenced by:  so3nr  4214
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