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Theorem potr 4787
Description: A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
potr  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )

Proof of Theorem potr
StepHypRef Expression
1 pocl 4782 . . 3  |-  ( R  Po  A  ->  (
( B  e.  A  /\  C  e.  A  /\  D  e.  A
)  ->  ( -.  B R B  /\  (
( B R C  /\  C R D )  ->  B R D ) ) ) )
21imp 430 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( -.  B R B  /\  ( ( B R C  /\  C R D )  ->  B R D ) ) )
32simprd 464 1  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    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:  po2nr  4788  po3nr  4789  pofun  4791  sotr  4797  poltletr  5252  predpo  5417  poxp  6919  frfi  7822  wemaplem2  8062  sornom  8705  zorn2lem7  8930  pospo  16170  pocnv  30191  poseq  30278  seqpo  31780
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