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Theorem potr 4517
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 4512 . . 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 420 . 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 451 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726   class class class wbr 4214    Po wpo 4503
This theorem is referenced by:  po2nr  4518  po3nr  4519  pofun  4521  sotr  4527  poltletr  5271  poxp  6460  frfi  7354  wemaplem2  7518  sornom  8159  zorn2lem7  8384  pospo  14432  pocnv  25389  predpo  25461  poseq  25530  seqpo  26453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4215  df-po 4505
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