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Theorem po2nr 4813
Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
po2nr  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )

Proof of Theorem po2nr
StepHypRef Expression
1 poirr 4811 . . 3  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 716 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 potr 4812 . . . . . 6  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  B  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
433exp2 1214 . . . . 5  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( C  e.  A  -> 
( B  e.  A  ->  ( ( B R C  /\  C R B )  ->  B R B ) ) ) ) )
54com34 83 . . . 4  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( B  e.  A  -> 
( C  e.  A  ->  ( ( B R C  /\  C R B )  ->  B R B ) ) ) ) )
65pm2.43d 48 . . 3  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( C  e.  A  -> 
( ( B R C  /\  C R B )  ->  B R B ) ) ) )
76imp32 433 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
82, 7mtod 177 1  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1767   class class class wbr 4447    Po wpo 4798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-po 4800
This theorem is referenced by:  po3nr  4814  so2nr  4824  soisoi  6212  wemaplem2  7972  pospo  15460
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