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Theorem clatpos 17715
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 2801 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2801 . . 3 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2801 . . 3 (glb‘𝐾) = (glb‘𝐾)
41, 2, 3isclat 17714 . 2 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
54simplbi 501 1 (𝐾 ∈ CLat → 𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  𝒫 cpw 4500  dom cdm 5523  cfv 6328  Basecbs 16478  Posetcpo 17545  lubclub 17547  glbcglb 17548  CLatccla 17712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-dm 5533  df-iota 6287  df-fv 6336  df-clat 17713
This theorem is referenced by: (None)
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