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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 32070 analog.) (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
atne0.z | β’ 0 = (0.βπΎ) |
atne0.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atn0 | β’ ((πΎ β AtLat β§ π β π΄) β π β 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | eqid 2724 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
3 | atne0.z | . . . 4 β’ 0 = (0.βπΎ) | |
4 | atne0.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | isat3 38671 | . . 3 β’ (πΎ β AtLat β (π β π΄ β (π β (BaseβπΎ) β§ π β 0 β§ βπ₯ β (BaseβπΎ)(π₯(leβπΎ)π β (π₯ = π β¨ π₯ = 0 ))))) |
6 | simp2 1134 | . . 3 β’ ((π β (BaseβπΎ) β§ π β 0 β§ βπ₯ β (BaseβπΎ)(π₯(leβπΎ)π β (π₯ = π β¨ π₯ = 0 ))) β π β 0 ) | |
7 | 5, 6 | syl6bi 253 | . 2 β’ (πΎ β AtLat β (π β π΄ β π β 0 )) |
8 | 7 | imp 406 | 1 β’ ((πΎ β AtLat β§ π β π΄) β π β 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β¨ wo 844 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 class class class wbr 5139 βcfv 6534 Basecbs 17145 lecple 17205 0.cp0 18380 Atomscatm 38627 AtLatcal 38628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-plt 18287 df-glb 18304 df-p0 18382 df-covers 38630 df-ats 38631 df-atl 38662 |
This theorem is referenced by: atncvrN 38679 atnle 38681 atlatmstc 38683 intnatN 38772 atcvrneN 38795 atcvrj2b 38797 2llnm3N 38934 pmapjat1 39218 lhpocnle 39381 lhpmatb 39396 lhp2atnle 39398 trlatn0 39537 ltrnnidn 39539 trlnidatb 39542 cdlemg33c 40073 cdlemg33e 40075 dihatexv 40703 |
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