| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > atnle0 | Structured version Visualization version GIF version | ||
| Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.) |
| Ref | Expression |
|---|---|
| atnle0.l | ⊢ ≤ = (le‘𝐾) |
| atnle0.z | ⊢ 0 = (0.‘𝐾) |
| atnle0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| Ref | Expression |
|---|---|
| atnle0 | ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | atlpos 39302 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Poset) |
| 3 | eqid 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 4 | atnle0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
| 5 | 3, 4 | atl0cl 39304 | . . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾)) |
| 6 | 5 | adantr 480 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ∈ (Base‘𝐾)) |
| 7 | atnle0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 8 | 3, 7 | atbase 39290 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
| 9 | 8 | adantl 481 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
| 10 | eqid 2737 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
| 11 | 4, 10, 7 | atcvr0 39289 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
| 12 | atnle0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 13 | 3, 12, 10 | cvrnle 39281 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 0 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑃) → ¬ 𝑃 ≤ 0 ) |
| 14 | 2, 6, 9, 11, 13 | syl31anc 1375 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 0.cp0 18468 ⋖ ccvr 39263 Atomscatm 39264 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-proset 18340 df-poset 18359 df-plt 18375 df-glb 18392 df-p0 18470 df-lat 18477 df-covers 39267 df-ats 39268 df-atl 39299 |
| This theorem is referenced by: pmap0 39767 trlnle 40188 cdlemg27b 40698 |
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