Theorem cvlposN 37341

Description: An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Assertion

Ref	Expression
cvlposN	⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Poset)

Proof of Theorem cvlposN

Step	Hyp	Ref	Expression
1		cvllat 37340	. 2 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
2		latpos 18156	. 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset)
3	1, 2	syl 17	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∈ wcel 2106  Posetcpo 18025  Latclat 18149  CvLatclc 37279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599  df-iota 6391  df-fv 6441  df-ov 7278  df-lat 18150  df-atl 37312  df-cvlat 37336

This theorem is referenced by: (None)