| 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 39301 | . 2 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) | |
| 2 | latpos 18483 | . 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 2108 Posetcpo 18353 Latclat 18476 AtLatcal 39265 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-lat 18477 df-atl 39299 |
| This theorem is referenced by: atlle0 39306 atnle0 39310 atlen0 39311 atcmp 39312 atcvreq0 39315 atlatmstc 39320 atlatle 39321 atlrelat1 39322 |
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