Theorem clatpos 18563

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 2762	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2762	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2762	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4	1, 2, 3	isclat 18562	. 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
5	4	simplbi 501	1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  𝒫 cpw 4561  dom cdm 5660  ‘cfv 6536  Basecbs 17275  Posetcpo 18369  lubclub 18371  glbcglb 18372  CLatccla 18560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-dm 5670  df-iota 6492  df-fv 6544  df-clat 18561

This theorem is used by: (None)