Theorem cvlposN 39819

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 39818	. 2 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
2		latpos 18395	. 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset)
3	1, 2	syl 17	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∈ wcel 2119  Posetcpo 18264  Latclat 18388  CvLatclc 39757

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-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-dm 5628  df-iota 6441  df-fv 6493  df-ov 7359  df-lat 18389  df-atl 39790  df-cvlat 39814

This theorem is referenced by: (None)