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Theorem potr 4091
Description: A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
potr ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))

Proof of Theorem potr
StepHypRef Expression
1 pocl 4086 . . 3 (𝑅 Po 𝐴 → ((𝐵𝐴𝐶𝐴𝐷𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
21imp 122 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷)))
32simprd 112 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  w3a 920  wcel 1434   class class class wbr 3805   Po wpo 4077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-po 4079
This theorem is referenced by:  po2nr  4092  po3nr  4093  pofun  4095  sotr  4101  issod  4102  poltletr  4775  poxp  5904
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