![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > atn0 | Structured version Visualization version GIF version |
Description: An atom is not zero. (atne0 32149 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 2728 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
2 | eqid 2728 | . . . 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 38774 | . . 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 1135 | . . 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 846 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2936 βwral 3057 class class class wbr 5143 βcfv 6543 Basecbs 17174 lecple 17234 0.cp0 18409 Atomscatm 38730 AtLatcal 38731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7371 df-plt 18316 df-glb 18333 df-p0 18411 df-covers 38733 df-ats 38734 df-atl 38765 |
This theorem is referenced by: atncvrN 38782 atnle 38784 atlatmstc 38786 intnatN 38875 atcvrneN 38898 atcvrj2b 38900 2llnm3N 39037 pmapjat1 39321 lhpocnle 39484 lhpmatb 39499 lhp2atnle 39501 trlatn0 39640 ltrnnidn 39642 trlnidatb 39645 cdlemg33c 40176 cdlemg33e 40178 dihatexv 40806 |
Copyright terms: Public domain | W3C validator |