Theorem potr 4429

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 4424	. . 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 124	. 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 114	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 104   /\ w3a 1005   e. wcel 2203   class class class wbr 4109   Po wpo 4415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-po 4417

This theorem is referenced by:  po2nr  4430  po3nr  4431  pofun  4433  sotr  4439  issod  4440  poltletr  5163  poxp  6428  fimax2gtrilemstep  7158