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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atlpos | Structured version Visualization version GIF version |
Description: An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
atlpos | ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atllat 39282 | . 2 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) | |
2 | latpos 18496 | . 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Posetcpo 18365 Latclat 18489 AtLatcal 39246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-iota 6516 df-fv 6571 df-lat 18490 df-atl 39280 |
This theorem is referenced by: atlle0 39287 atnle0 39291 atlen0 39292 atcmp 39293 atcvreq0 39296 atlatmstc 39301 atlatle 39302 atlrelat1 39303 |
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