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Mirrors > Home > MPE Home > Th. List > lubel | Structured version Visualization version GIF version |
Description: An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
lublem.b | β’ π΅ = (BaseβπΎ) |
lublem.l | β’ β€ = (leβπΎ) |
lublem.u | β’ π = (lubβπΎ) |
Ref | Expression |
---|---|
lubel | β’ ((πΎ β CLat β§ π β π β§ π β π΅) β π β€ (πβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatl 18465 | . . . 4 β’ (πΎ β CLat β πΎ β Lat) | |
2 | ssel 3975 | . . . . 5 β’ (π β π΅ β (π β π β π β π΅)) | |
3 | 2 | impcom 408 | . . . 4 β’ ((π β π β§ π β π΅) β π β π΅) |
4 | lublem.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
5 | lublem.u | . . . . 5 β’ π = (lubβπΎ) | |
6 | 4, 5 | lubsn 18439 | . . . 4 β’ ((πΎ β Lat β§ π β π΅) β (πβ{π}) = π) |
7 | 1, 3, 6 | syl2an 596 | . . 3 β’ ((πΎ β CLat β§ (π β π β§ π β π΅)) β (πβ{π}) = π) |
8 | 7 | 3impb 1115 | . 2 β’ ((πΎ β CLat β§ π β π β§ π β π΅) β (πβ{π}) = π) |
9 | snssi 4811 | . . . 4 β’ (π β π β {π} β π) | |
10 | lublem.l | . . . . 5 β’ β€ = (leβπΎ) | |
11 | 4, 10, 5 | lubss 18470 | . . . 4 β’ ((πΎ β CLat β§ π β π΅ β§ {π} β π) β (πβ{π}) β€ (πβπ)) |
12 | 9, 11 | syl3an3 1165 | . . 3 β’ ((πΎ β CLat β§ π β π΅ β§ π β π) β (πβ{π}) β€ (πβπ)) |
13 | 12 | 3com23 1126 | . 2 β’ ((πΎ β CLat β§ π β π β§ π β π΅) β (πβ{π}) β€ (πβπ)) |
14 | 8, 13 | eqbrtrrd 5172 | 1 β’ ((πΎ β CLat β§ π β π β§ π β π΅) β π β€ (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wss 3948 {csn 4628 class class class wbr 5148 βcfv 6543 Basecbs 17148 lecple 17208 lubclub 18266 Latclat 18388 CLatccla 18455 |
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-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 18252 df-poset 18270 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-lat 18389 df-clat 18456 |
This theorem is referenced by: lubun 18472 atlatmstc 38492 2polssN 39089 |
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