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| Mirrors > Home > MPE Home > Th. List > Mathboxes > atn0 | Structured version Visualization version GIF version | ||
| Description: An atom is not zero. (atne0 32364 analog.) (Contributed by NM, 5-Nov-2012.) |
| Ref | Expression |
|---|---|
| atne0.z | ⊢ 0 = (0.‘𝐾) |
| atne0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| Ref | Expression |
|---|---|
| atn0 | ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 2 | eqid 2737 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 3 | atne0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
| 4 | atne0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 5 | 1, 2, 3, 4 | isat3 39308 | . . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))) |
| 6 | simp2 1138 | . . 3 ⊢ ((𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) → 𝑃 ≠ 0 ) | |
| 7 | 5, 6 | biimtrdi 253 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 → 𝑃 ≠ 0 )) |
| 8 | 7 | imp 406 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 0.cp0 18468 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-plt 18375 df-glb 18392 df-p0 18470 df-covers 39267 df-ats 39268 df-atl 39299 |
| This theorem is referenced by: atncvrN 39316 atnle 39318 atlatmstc 39320 intnatN 39409 atcvrneN 39432 atcvrj2b 39434 2llnm3N 39571 pmapjat1 39855 lhpocnle 40018 lhpmatb 40033 lhp2atnle 40035 trlatn0 40174 ltrnnidn 40176 trlnidatb 40179 cdlemg33c 40710 cdlemg33e 40712 dihatexv 41340 |
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