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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atnle0 | Structured version Visualization version GIF version |
Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.) |
Ref | Expression |
---|---|
atnle0.l | β’ β€ = (leβπΎ) |
atnle0.z | β’ 0 = (0.βπΎ) |
atnle0.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atnle0 | β’ ((πΎ β AtLat β§ π β π΄) β Β¬ π β€ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atlpos 38773 | . . 3 β’ (πΎ β AtLat β πΎ β Poset) | |
2 | 1 | adantr 480 | . 2 β’ ((πΎ β AtLat β§ π β π΄) β πΎ β Poset) |
3 | eqid 2728 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | atnle0.z | . . . 4 β’ 0 = (0.βπΎ) | |
5 | 3, 4 | atl0cl 38775 | . . 3 β’ (πΎ β AtLat β 0 β (BaseβπΎ)) |
6 | 5 | adantr 480 | . 2 β’ ((πΎ β AtLat β§ π β π΄) β 0 β (BaseβπΎ)) |
7 | atnle0.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
8 | 3, 7 | atbase 38761 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
9 | 8 | adantl 481 | . 2 β’ ((πΎ β AtLat β§ π β π΄) β π β (BaseβπΎ)) |
10 | eqid 2728 | . . 3 β’ ( β βπΎ) = ( β βπΎ) | |
11 | 4, 10, 7 | atcvr0 38760 | . 2 β’ ((πΎ β AtLat β§ π β π΄) β 0 ( β βπΎ)π) |
12 | atnle0.l | . . 3 β’ β€ = (leβπΎ) | |
13 | 3, 12, 10 | cvrnle 38752 | . 2 β’ (((πΎ β Poset β§ 0 β (BaseβπΎ) β§ π β (BaseβπΎ)) β§ 0 ( β βπΎ)π) β Β¬ π β€ 0 ) |
14 | 2, 6, 9, 11, 13 | syl31anc 1371 | 1 β’ ((πΎ β AtLat β§ π β π΄) β Β¬ π β€ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 Basecbs 17180 lecple 17240 Posetcpo 18299 0.cp0 18415 β ccvr 38734 Atomscatm 38735 AtLatcal 38736 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-proset 18287 df-poset 18305 df-plt 18322 df-glb 18339 df-p0 18417 df-lat 18424 df-covers 38738 df-ats 38739 df-atl 38770 |
This theorem is referenced by: pmap0 39238 trlnle 39659 cdlemg27b 40169 |
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