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

Theorem sotr 5603
Description: A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)
Assertion
Ref Expression
sotr ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))

Proof of Theorem sotr
StepHypRef Expression
1 sopo 5598 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 potr 5592 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
31, 2sylan 579 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wcel 2098   class class class wbr 5139   Po wpo 5577   Or wor 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-po 5579  df-so 5580
This theorem is referenced by:  sotr2  5611  sotr3  5618  wetrep  5660  wereu2  5664  sotri  6119  suplub2  9453  slttr  27599  sotrd  35231  fin2solem  36968
  Copyright terms: Public domain W3C validator