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Theorem postr 17268
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 17265 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
43simp3d 1175 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157   class class class wbr 4843  cfv 6101  Basecbs 16184  lecple 16274  Posetcpo 17255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-poset 17261
This theorem is referenced by:  plttr  17285  joinle  17329  meetle  17343  lattr  17371  odupos  17450  omndadd2d  30224  omndadd2rd  30225  omndmul2  30228  atlatle  35341  cvratlem  35442  llncmp  35543  llncvrlpln  35579  lplncmp  35583  lplncvrlvol  35637  lvolcmp  35638  pmaple  35782
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