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Theorem atlpos 39283
Description: An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
atlpos (𝐾 ∈ AtLat → 𝐾 ∈ Poset)

Proof of Theorem atlpos
StepHypRef Expression
1 atllat 39282 . 2 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
2 latpos 18496 . 2 (𝐾 ∈ Lat → 𝐾 ∈ Poset)
31, 2syl 17 1 (𝐾 ∈ AtLat → 𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Posetcpo 18365  Latclat 18489  AtLatcal 39246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-lat 18490  df-atl 39280
This theorem is referenced by:  atlle0  39287  atnle0  39291  atlen0  39292  atcmp  39293  atcvreq0  39296  atlatmstc  39301  atlatle  39302  atlrelat1  39303
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