Theorem clatpos 18458

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 2739	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2739	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2739	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4	1, 2, 3	isclat 18457	. 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 396   = wceq 1547   ∈ wcel 2119  𝒫 cpw 4529  dom cdm 5618  ‘cfv 6485  Basecbs 17170  Posetcpo 18264  lubclub 18266  glbcglb 18267  CLatccla 18455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-dm 5628  df-iota 6441  df-fv 6493  df-clat 18456

This theorem is referenced by: (None)