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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > polssatN | Structured version Visualization version GIF version |
Description: The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polssat.a | β’ π΄ = (AtomsβπΎ) |
polssat.p | β’ β₯ = (β₯πβπΎ) |
Ref | Expression |
---|---|
polssatN | β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | polssat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
2 | eqid 2733 | . . 3 β’ (PSubSpβπΎ) = (PSubSpβπΎ) | |
3 | polssat.p | . . 3 β’ β₯ = (β₯πβπΎ) | |
4 | 1, 2, 3 | polsubN 38778 | . 2 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) β (PSubSpβπΎ)) |
5 | 1, 2 | psubssat 38625 | . 2 β’ ((πΎ β HL β§ ( β₯ βπ) β (PSubSpβπΎ)) β ( β₯ βπ) β π΄) |
6 | 4, 5 | syldan 592 | 1 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3949 βcfv 6544 Atomscatm 38133 HLchlt 38220 PSubSpcpsubsp 38367 β₯πcpolN 38773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-proset 18248 df-poset 18266 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p1 18379 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-psubsp 38374 df-pmap 38375 df-polarityN 38774 |
This theorem is referenced by: 2polcon4bN 38789 polcon2N 38790 pclss2polN 38792 2pmaplubN 38797 paddunN 38798 ispsubcl2N 38818 poml5N 38825 osumcllem1N 38827 osumcllem2N 38828 osumcllem3N 38829 osumcllem9N 38835 osumcllem11N 38837 pexmidN 38840 pexmidlem2N 38842 pexmidlem3N 38843 pexmidlem7N 38847 pexmidlem8N 38848 |
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