Theorem clatpos 18415

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

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2733	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2733	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4	1, 2, 3	isclat 18414	. 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
5	4	simplbi 497	1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  𝒫 cpw 4551  dom cdm 5621  ‘cfv 6489  Basecbs 17127  Posetcpo 18221  lubclub 18223  glbcglb 18224  CLatccla 18412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-dm 5631  df-iota 6445  df-fv 6497  df-clat 18413

This theorem is referenced by: (None)