ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  potr Unicode version

Theorem potr 4126
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 4121 . . 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 122 . 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 112 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 102    /\ w3a 924    e. wcel 1438   class class class wbr 3837    Po wpo 4112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-po 4114
This theorem is referenced by:  po2nr  4127  po3nr  4128  pofun  4130  sotr  4136  issod  4137  poltletr  4819  poxp  5979  fimax2gtrilemstep  6596
  Copyright terms: Public domain W3C validator