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Mirrors > Home > ILE Home > Th. List > bndndx | Unicode version |
Description: A bounded real sequence is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.) |
Ref | Expression |
---|---|
bndndx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arch 8974 | . . . 4 | |
2 | nnre 8727 | . . . . . 6 | |
3 | lelttr 7852 | . . . . . . . . . . 11 | |
4 | ltle 7851 | . . . . . . . . . . . 12 | |
5 | 4 | 3adant2 1000 | . . . . . . . . . . 11 |
6 | 3, 5 | syld 45 | . . . . . . . . . 10 |
7 | 6 | exp5o 1204 | . . . . . . . . 9 |
8 | 7 | com3l 81 | . . . . . . . 8 |
9 | 8 | imp4b 347 | . . . . . . 7 |
10 | 9 | com23 78 | . . . . . 6 |
11 | 2, 10 | sylan2 284 | . . . . 5 |
12 | 11 | reximdva 2534 | . . . 4 |
13 | 1, 12 | mpd 13 | . . 3 |
14 | r19.35-1 2581 | . . 3 | |
15 | 13, 14 | syl 14 | . 2 |
16 | 15 | rexlimiv 2543 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wcel 1480 wral 2416 wrex 2417 class class class wbr 3929 cr 7619 clt 7800 cle 7801 cn 8720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-inn 8721 |
This theorem is referenced by: (None) |
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