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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 31593 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 2732 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | eqid 2732 | . . . 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 38172 | . . 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 1137 | . . 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 252 | . 2 β’ (πΎ β AtLat β (π β π΄ β π β 0 )) |
8 | 7 | imp 407 | 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 396 β¨ wo 845 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 class class class wbr 5148 βcfv 6543 Basecbs 17143 lecple 17203 0.cp0 18375 Atomscatm 38128 AtLatcal 38129 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-plt 18282 df-glb 18299 df-p0 18377 df-covers 38131 df-ats 38132 df-atl 38163 |
This theorem is referenced by: atncvrN 38180 atnle 38182 atlatmstc 38184 intnatN 38273 atcvrneN 38296 atcvrj2b 38298 2llnm3N 38435 pmapjat1 38719 lhpocnle 38882 lhpmatb 38897 lhp2atnle 38899 trlatn0 39038 ltrnnidn 39040 trlnidatb 39043 cdlemg33c 39574 cdlemg33e 39576 dihatexv 40204 |
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