Theorem clatpos 18219

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 18218	. 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
5	4	simplbi 498	1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  𝒫 cpw 4533  dom cdm 5589  ‘cfv 6433  Basecbs 16912  Posetcpo 18025  lubclub 18027  glbcglb 18028  CLatccla 18216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-dm 5599  df-iota 6391  df-fv 6441  df-clat 18217

This theorem is referenced by: (None)