Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  clatpos Structured version   Visualization version   GIF version

Theorem clatpos 17091
 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
StepHypRef Expression
1 eqid 2620 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2620 . . 3 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2620 . . 3 (glb‘𝐾) = (glb‘𝐾)
41, 2, 3isclat 17090 . 2 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾))))
54simplbi 476 1 (𝐾 ∈ CLat → 𝐾 ∈ Poset)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  𝒫 cpw 4149  dom cdm 5104  ‘cfv 5876  Basecbs 15838  Posetcpo 16921  lubclub 16923  glbcglb 16924  CLatccla 17088 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-dm 5114  df-iota 5839  df-fv 5884  df-clat 17089 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator