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Theorem postr 18366
Description: A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)
Hypotheses
Ref Expression
posi.b 𝐵 = (Base‘𝐾)
posi.l = (le‘𝐾)
Assertion
Ref Expression
postr ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))

Proof of Theorem postr
StepHypRef Expression
1 posi.b . . 3 𝐵 = (Base‘𝐾)
2 posi.l . . 3 = (le‘𝐾)
31, 2posi 18363 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
43simp3d 1145 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  Posetcpo 18353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-poset 18359
This theorem is referenced by:  odupos  18373  plttr  18387  joinle  18431  meetle  18445  lattr  18489  omndadd2d  33085  omndadd2rd  33086  omndmul2  33089  atlatle  39321  cvratlem  39423  llncmp  39524  llncvrlpln  39560  lplncmp  39564  lplncvrlvol  39618  lvolcmp  39619  pmaple  39763
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