Theorem clatpos 18675

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 2760	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2760	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2760	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4	1, 2, 3	isclat 18674	. 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 1569   ∈ wcel 2144  𝒫 cpw 4556  dom cdm 5650  ‘cfv 6538  Basecbs 17387  Posetcpo 18481  lubclub 18483  glbcglb 18484  CLatccla 18672

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-dm 5660  df-iota 6494  df-fv 6546  df-clat 18673

This theorem is used by: (None)