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Theorem cvllat 36622
Description: An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
cvllat (𝐾 ∈ CvLat → 𝐾 ∈ Lat)

Proof of Theorem cvllat
StepHypRef Expression
1 cvlatl 36621 . 2 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
2 atllat 36596 . 2 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
31, 2syl 17 1 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2111  Latclat 17647  AtLatcal 36560  CvLatclc 36561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-dm 5529  df-iota 6283  df-fv 6332  df-ov 7138  df-atl 36594  df-cvlat 36618
This theorem is referenced by:  cvlposN  36623  cvlexch2  36625  cvlexchb1  36626  cvlexchb2  36627  cvlatexchb2  36631  cvlatexch1  36632  cvlatexch2  36633  cvlatexch3  36634  cvlcvr1  36635  cvlcvrp  36636  cvlatcvr2  36638  cvlsupr2  36639  cvlsupr7  36644  cvlsupr8  36645
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