MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  po2nr Structured version   Unicode version

Theorem po2nr 4788
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 4786 . . 3  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
21adantrr 721 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  -.  B R B )
3 potr 4787 . . . . . 6  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  B  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
433exp2 1223 . . . . 5  |-  ( R  Po  A  ->  ( B  e.  A  ->  ( C  e.  A  -> 
( B  e.  A  ->  ( ( B R C  /\  C R B )  ->  B R B ) ) ) ) )
54com34 86 . . . 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 434 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B R C  /\  C R B )  ->  B R B ) )
82, 7mtod 180 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 370    e. wcel 1870   class class class wbr 4426    Po wpo 4773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-po 4775
This theorem is referenced by:  po3nr  4789  so2nr  4799  soisoi  6234  wemaplem2  8062  pospo  16170
  Copyright terms: Public domain W3C validator