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Theorem po2nr 4074
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 4072 . . 3  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 456 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 potr 4073 . . . . . 6  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  B  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
433exp2 1133 . . . . 5  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( C  e.  A  -> 
( B  e.  A  ->  ( ( B R C  /\  C R B )  ->  B R B ) ) ) ) )
54com34 81 . . . 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 248 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
82, 7mtod 599 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 101    e. wcel 1409   class class class wbr 3792    Po wpo 4059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-po 4061
This theorem is referenced by:  po3nr  4075  so2nr  4086
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