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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 30049 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 2818 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2818 | . . . 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 36323 | . . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))) |
6 | simp2 1129 | . . 3 ⊢ ((𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) → 𝑃 ≠ 0 ) | |
7 | 5, 6 | syl6bi 254 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 → 𝑃 ≠ 0 )) |
8 | 7 | imp 407 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 lecple 16560 0.cp0 17635 Atomscatm 36279 AtLatcal 36280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-plt 17556 df-glb 17573 df-p0 17637 df-covers 36282 df-ats 36283 df-atl 36314 |
This theorem is referenced by: atncvrN 36331 atnle 36333 atlatmstc 36335 intnatN 36423 atcvrneN 36446 atcvrj2b 36448 2llnm3N 36585 pmapjat1 36869 lhpocnle 37032 lhpmatb 37047 lhp2atnle 37049 trlatn0 37188 ltrnnidn 37190 trlnidatb 37193 cdlemg33c 37724 cdlemg33e 37726 dihatexv 38354 |
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