MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sotr Structured version   Unicode version

Theorem sotr 4811
Description: A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)
Assertion
Ref Expression
sotr  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )

Proof of Theorem sotr
StepHypRef Expression
1 sopo 4806 . 2  |-  ( R  Or  A  ->  R  Po  A )
2 potr 4801 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
31, 2sylan 469 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    e. wcel 1823   class class class wbr 4439    Po wpo 4787    Or wor 4788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-po 4789  df-so 4790
This theorem is referenced by:  sotr2  4818  wetrep  4861  wereu2  4865  sotri  5382  suplub2  7912  slttr  29674  fin2solem  30282
  Copyright terms: Public domain W3C validator