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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 32374 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 2735 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2735 | . . . 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 39289 | . . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))) |
6 | simp2 1136 | . . 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 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 0.cp0 18481 Atomscatm 39245 AtLatcal 39246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-plt 18388 df-glb 18405 df-p0 18483 df-covers 39248 df-ats 39249 df-atl 39280 |
This theorem is referenced by: atncvrN 39297 atnle 39299 atlatmstc 39301 intnatN 39390 atcvrneN 39413 atcvrj2b 39415 2llnm3N 39552 pmapjat1 39836 lhpocnle 39999 lhpmatb 40014 lhp2atnle 40016 trlatn0 40155 ltrnnidn 40157 trlnidatb 40160 cdlemg33c 40691 cdlemg33e 40693 dihatexv 41321 |
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