Theorem potr 4230

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

Step	Hyp	Ref	Expression
1		pocl 4225	. . 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 ) ) ) )
2	1	imp 123	. 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 ) ) )
3	2	simprd 113	1 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( B R C /\ C R D ) -> B R D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 962   e. wcel 1480   class class class wbr 3929   Po wpo 4216

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-po 4218

This theorem is referenced by:  po2nr  4231  po3nr  4232  pofun  4234  sotr  4240  issod  4241  poltletr  4939  poxp  6129  fimax2gtrilemstep  6794