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Mathbox for Norm Megill |
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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 29780 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 2778 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2778 | . . . 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 35466 | . . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))) |
6 | simp2 1128 | . . 3 ⊢ ((𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) → 𝑃 ≠ 0 ) | |
7 | 5, 6 | syl6bi 245 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 → 𝑃 ≠ 0 )) |
8 | 7 | imp 397 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 class class class wbr 4888 ‘cfv 6137 Basecbs 16259 lecple 16349 0.cp0 17427 Atomscatm 35422 AtLatcal 35423 |
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-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
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-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-plt 17348 df-glb 17365 df-p0 17429 df-covers 35425 df-ats 35426 df-atl 35457 |
This theorem is referenced by: atncvrN 35474 atnle 35476 atlatmstc 35478 intnatN 35566 atcvrneN 35589 atcvrj2b 35591 2llnm3N 35728 pmapjat1 36012 lhpocnle 36175 lhpmatb 36190 lhp2atnle 36192 trlatn0 36331 ltrnnidn 36333 trlnidatb 36336 cdlemg33c 36867 cdlemg33e 36869 dihatexv 37497 |
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