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Theorem po3nr 4293
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 4292 . . 3  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  D  e.  A
) )  ->  -.  ( B R D  /\  D R B ) )
213adantr2 1152 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  -.  ( B R D  /\  D R B ) )
3 df-3an 975 . . 3  |-  ( ( B R C  /\  C R D  /\  D R B )  <->  ( ( B R C  /\  C R D )  /\  D R B ) )
4 potr 4291 . . . 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 658 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 973    e. wcel 2141   class class class wbr 3987    Po wpo 4277
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-po 4279
This theorem is referenced by:  so3nr  4305
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