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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 30128 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 2798 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2798 | . . . 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 36603 | . . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))) |
6 | simp2 1134 | . . 3 ⊢ ((𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))) → 𝑃 ≠ 0 ) | |
7 | 5, 6 | syl6bi 256 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 → 𝑃 ≠ 0 )) |
8 | 7 | imp 410 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 0.cp0 17639 Atomscatm 36559 AtLatcal 36560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-plt 17560 df-glb 17577 df-p0 17641 df-covers 36562 df-ats 36563 df-atl 36594 |
This theorem is referenced by: atncvrN 36611 atnle 36613 atlatmstc 36615 intnatN 36703 atcvrneN 36726 atcvrj2b 36728 2llnm3N 36865 pmapjat1 37149 lhpocnle 37312 lhpmatb 37327 lhp2atnle 37329 trlatn0 37468 ltrnnidn 37470 trlnidatb 37473 cdlemg33c 38004 cdlemg33e 38006 dihatexv 38634 |
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