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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 35454 | . 2 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) | |
2 | latpos 17436 | . 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 2107 Posetcpo 17326 Latclat 17431 AtLatcal 35418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-dm 5365 df-iota 6099 df-fv 6143 df-lat 17432 df-atl 35452 |
This theorem is referenced by: atlle0 35459 atnle0 35463 atlen0 35464 atcmp 35465 atcvreq0 35468 atlatmstc 35473 atlatle 35474 atlrelat1 35475 |
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