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Theorem po2nr 4519
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 4517 . . 3  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 699 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 potr 4518 . . . . . 6  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  B  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
433exp2 1172 . . . . 5  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( C  e.  A  -> 
( B  e.  A  ->  ( ( B R C  /\  C R B )  ->  B R B ) ) ) ) )
54com34 80 . . . 4  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( B  e.  A  -> 
( C  e.  A  ->  ( ( B R C  /\  C R B )  ->  B R B ) ) ) ) )
65pm2.43d 47 . . 3  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( C  e.  A  -> 
( ( B R C  /\  C R B )  ->  B R B ) ) ) )
76imp32 424 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
82, 7mtod 171 1  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  ( B R C  /\  C R B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    e. wcel 1726   class class class wbr 4215    Po wpo 4504
This theorem is referenced by:  po3nr  4520  so2nr  4530  soisoi  6051  wemaplem2  7519  pospo  14435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-po 4506
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