Theorem cvlposN 38661

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 38660	. 2 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
2		latpos 18401	. 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset)
3	1, 2	syl 17	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∈ wcel 2105  Posetcpo 18270  Latclat 18394  CvLatclc 38599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-iota 6495  df-fv 6551  df-ov 7415  df-lat 18395  df-atl 38632  df-cvlat 38656

This theorem is referenced by: (None)