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Theorem sotr 5055
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 5050 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 potr 5045 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
31, 2sylan 488 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037  wcel 1989   class class class wbr 4651   Po wpo 5031   Or wor 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-po 5033  df-so 5034
This theorem is referenced by:  sotr2  5062  wetrep  5105  wereu2  5109  sotri  5521  suplub2  8364  sotrd  31641  sotr3  31642  slttr  31856  fin2solem  33375
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