Theorem potr 4045

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 4040	. . 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 115	. 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 107	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 97   /\ w3a 885   e. wcel 1393   class class class wbr 3764   Po wpo 4031

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033

This theorem is referenced by:  po2nr  4046  po3nr  4047  pofun  4049  sotr  4055  issod  4056  poltletr  4725  poxp  5853