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Theorem po2nr 4292
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 4290 . . 3  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 476 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 potr 4291 . . . . . 6  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  B  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
433exp2 1220 . . . . 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 50 . . 3  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( C  e.  A  -> 
( ( B R C  /\  C R B )  ->  B R B ) ) ) )
76imp32 255 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
82, 7mtod 658 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 103    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:  po3nr  4293  so2nr  4304  tridc  6873
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