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Theorem cvllat 39269
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 39268 . 2 (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
2 atllat 39243 . 2 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
31, 2syl 17 1 (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  Latclat 18477  AtLatcal 39207  CvLatclc 39208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-dm 5693  df-iota 6510  df-fv 6566  df-ov 7428  df-atl 39241  df-cvlat 39265
This theorem is referenced by:  cvlposN  39270  cvlexch2  39272  cvlexchb1  39273  cvlexchb2  39274  cvlatexchb2  39278  cvlatexch1  39279  cvlatexch2  39280  cvlatexch3  39281  cvlcvr1  39282  cvlcvrp  39283  cvlatcvr2  39285  cvlsupr2  39286  cvlsupr7  39291  cvlsupr8  39292
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