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Theorem clatpos 16933
Description: A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.)
Assertion
Ref Expression
clatpos (𝐾 ∈ CLat → 𝐾 ∈ Poset)

Proof of Theorem clatpos
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2610 . . 3 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2610 . . 3 (glb‘𝐾) = (glb‘𝐾)
41, 2, 3isclat 16932 . 2 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
54simplbi 475 1 (𝐾 ∈ CLat → 𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  𝒫 cpw 4108  dom cdm 5038  cfv 5804  Basecbs 15695  Posetcpo 16763  lubclub 16765  glbcglb 16766  CLatccla 16930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-clat 16931
This theorem is referenced by: (None)
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