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Theorem sotr 5284
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 5279 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 potr 5274 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
31, 2sylan 577 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113  wcel 2166   class class class wbr 4872   Po wpo 5260   Or wor 5261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-po 5262  df-so 5263
This theorem is referenced by:  sotr2  5291  wetrep  5334  wereu2  5338  sotri  5764  suplub2  8635  sotrd  32196  sotr3  32197  slttr  32410  fin2solem  33937
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