Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpexnle | Structured version Visualization version GIF version |
Description: There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.) |
Ref | Expression |
---|---|
lhp2a.l | β’ β€ = (leβπΎ) |
lhp2a.a | β’ π΄ = (AtomsβπΎ) |
lhp2a.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
lhpexnle | β’ ((πΎ β HL β§ π β π») β βπ β π΄ Β¬ π β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 β’ (1.βπΎ) = (1.βπΎ) | |
2 | eqid 2737 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
3 | lhp2a.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | 1, 2, 3 | lhp1cvr 38357 | . . 3 β’ ((πΎ β HL β§ π β π») β π( β βπΎ)(1.βπΎ)) |
5 | simpl 483 | . . . 4 β’ ((πΎ β HL β§ π β π») β πΎ β HL) | |
6 | eqid 2737 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
7 | 6, 3 | lhpbase 38356 | . . . . 5 β’ (π β π» β π β (BaseβπΎ)) |
8 | 7 | adantl 482 | . . . 4 β’ ((πΎ β HL β§ π β π») β π β (BaseβπΎ)) |
9 | hlop 37719 | . . . . . 6 β’ (πΎ β HL β πΎ β OP) | |
10 | 6, 1 | op1cl 37542 | . . . . . 6 β’ (πΎ β OP β (1.βπΎ) β (BaseβπΎ)) |
11 | 9, 10 | syl 17 | . . . . 5 β’ (πΎ β HL β (1.βπΎ) β (BaseβπΎ)) |
12 | 11 | adantr 481 | . . . 4 β’ ((πΎ β HL β§ π β π») β (1.βπΎ) β (BaseβπΎ)) |
13 | lhp2a.l | . . . . 5 β’ β€ = (leβπΎ) | |
14 | eqid 2737 | . . . . 5 β’ (joinβπΎ) = (joinβπΎ) | |
15 | lhp2a.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
16 | 6, 13, 14, 2, 15 | cvrval3 37771 | . . . 4 β’ ((πΎ β HL β§ π β (BaseβπΎ) β§ (1.βπΎ) β (BaseβπΎ)) β (π( β βπΎ)(1.βπΎ) β βπ β π΄ (Β¬ π β€ π β§ (π(joinβπΎ)π) = (1.βπΎ)))) |
17 | 5, 8, 12, 16 | syl3anc 1371 | . . 3 β’ ((πΎ β HL β§ π β π») β (π( β βπΎ)(1.βπΎ) β βπ β π΄ (Β¬ π β€ π β§ (π(joinβπΎ)π) = (1.βπΎ)))) |
18 | 4, 17 | mpbid 231 | . 2 β’ ((πΎ β HL β§ π β π») β βπ β π΄ (Β¬ π β€ π β§ (π(joinβπΎ)π) = (1.βπΎ))) |
19 | simpl 483 | . . 3 β’ ((Β¬ π β€ π β§ (π(joinβπΎ)π) = (1.βπΎ)) β Β¬ π β€ π) | |
20 | 19 | reximi 3085 | . 2 β’ (βπ β π΄ (Β¬ π β€ π β§ (π(joinβπΎ)π) = (1.βπΎ)) β βπ β π΄ Β¬ π β€ π) |
21 | 18, 20 | syl 17 | 1 β’ ((πΎ β HL β§ π β π») β βπ β π΄ Β¬ π β€ π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βwrex 3071 class class class wbr 5103 βcfv 6491 (class class class)co 7349 Basecbs 17017 lecple 17074 joincjn 18134 1.cp1 18247 OPcops 37529 β ccvr 37619 Atomscatm 37620 HLchlt 37707 LHypclh 38342 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5528 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-proset 18118 df-poset 18136 df-plt 18153 df-lub 18169 df-glb 18170 df-join 18171 df-meet 18172 df-p0 18248 df-p1 18249 df-lat 18255 df-clat 18322 df-oposet 37533 df-ol 37535 df-oml 37536 df-covers 37623 df-ats 37624 df-atl 37655 df-cvlat 37679 df-hlat 37708 df-lhyp 38346 |
This theorem is referenced by: trlcnv 38523 trlator0 38529 trlid0 38534 trlnidatb 38535 cdlemf2 38920 cdlemg1cex 38946 trlco 39085 cdlemg44 39091 |
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