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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 2732 | . . 3 β’ (PSubSpβπΎ) = (PSubSpβπΎ) | |
3 | polssat.p | . . 3 β’ β₯ = (β₯πβπΎ) | |
4 | 1, 2, 3 | polsubN 38864 | . 2 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) β (PSubSpβπΎ)) |
5 | 1, 2 | psubssat 38711 | . 2 β’ ((πΎ β HL β§ ( β₯ βπ) β (PSubSpβπΎ)) β ( β₯ βπ) β π΄) |
6 | 4, 5 | syldan 591 | 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 396 = wceq 1541 β wcel 2106 β wss 3948 βcfv 6543 Atomscatm 38219 HLchlt 38306 PSubSpcpsubsp 38453 β₯πcpolN 38859 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7367 df-ov 7414 df-oprab 7415 df-proset 18250 df-poset 18268 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p1 18381 df-lat 18387 df-clat 18454 df-oposet 38132 df-ol 38134 df-oml 38135 df-ats 38223 df-atl 38254 df-cvlat 38278 df-hlat 38307 df-psubsp 38460 df-pmap 38461 df-polarityN 38860 |
This theorem is referenced by: 2polcon4bN 38875 polcon2N 38876 pclss2polN 38878 2pmaplubN 38883 paddunN 38884 ispsubcl2N 38904 poml5N 38911 osumcllem1N 38913 osumcllem2N 38914 osumcllem3N 38915 osumcllem9N 38921 osumcllem11N 38923 pexmidN 38926 pexmidlem2N 38928 pexmidlem3N 38929 pexmidlem7N 38933 pexmidlem8N 38934 |
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