Theorem clatpos 18134

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 2738	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2738	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2738	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4	1, 2, 3	isclat 18133	. 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 1539   ∈ wcel 2108  𝒫 cpw 4530  dom cdm 5580  ‘cfv 6418  Basecbs 16840  Posetcpo 17940  lubclub 17942  glbcglb 17943  CLatccla 18131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-dm 5590  df-iota 6376  df-fv 6426  df-clat 18132

This theorem is referenced by: (None)