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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2atneat | Structured version Visualization version GIF version |
Description: The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.) |
Ref | Expression |
---|---|
2atneat.j | β’ β¨ = (joinβπΎ) |
2atneat.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
2atneat | β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π)) β Β¬ (π β¨ π) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π)) β πΎ β HL) | |
2 | simpr1 1191 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π)) β π β π΄) | |
3 | simpr2 1192 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π)) β π β π΄) | |
4 | simpr3 1193 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π)) β π β π) | |
5 | 2atneat.j | . . . 4 β’ β¨ = (joinβπΎ) | |
6 | 2atneat.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
7 | eqid 2726 | . . . 4 β’ (LLinesβπΎ) = (LLinesβπΎ) | |
8 | 5, 6, 7 | llni2 38895 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β π) β (π β¨ π) β (LLinesβπΎ)) |
9 | 1, 2, 3, 4, 8 | syl31anc 1370 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π)) β (π β¨ π) β (LLinesβπΎ)) |
10 | 6, 7 | llnneat 38897 | . 2 β’ ((πΎ β HL β§ (π β¨ π) β (LLinesβπΎ)) β Β¬ (π β¨ π) β π΄) |
11 | 9, 10 | syldan 590 | 1 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π)) β Β¬ (π β¨ π) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βcfv 6536 (class class class)co 7404 joincjn 18273 Atomscatm 38645 HLchlt 38732 LLinesclln 38874 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 |
This theorem is referenced by: cdleme18b 39675 |
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