Theorem clatpos 18701

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 2763	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2763	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2763	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4	1, 2, 3	isclat 18700	. 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
5	4	simplbi 502	1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  𝒫 cpw 4560  dom cdm 5655  ‘cfv 6555  Basecbs 17413  Posetcpo 18507  lubclub 18509  glbcglb 18510  CLatccla 18698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-dm 5665  df-iota 6509  df-fv 6563  df-clat 18699

This theorem is used by: (None)