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Mirrors > Home > ILE Home > Th. List > eluzuzle | Unicode version |
Description: An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.) |
Ref | Expression |
---|---|
eluzuzle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 9332 | . 2 | |
2 | simpll 518 | . . . 4 | |
3 | simpr2 988 | . . . 4 | |
4 | zre 9058 | . . . . . 6 | |
5 | 4 | ad2antrr 479 | . . . . 5 |
6 | zre 9058 | . . . . . . 7 | |
7 | 6 | 3ad2ant1 1002 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | zre 9058 | . . . . . . 7 | |
10 | 9 | 3ad2ant2 1003 | . . . . . 6 |
11 | 10 | adantl 275 | . . . . 5 |
12 | simplr 519 | . . . . 5 | |
13 | simpr3 989 | . . . . 5 | |
14 | 5, 8, 11, 12, 13 | letrd 7886 | . . . 4 |
15 | eluz2 9332 | . . . 4 | |
16 | 2, 3, 14, 15 | syl3anbrc 1165 | . . 3 |
17 | 16 | ex 114 | . 2 |
18 | 1, 17 | syl5bi 151 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wcel 1480 class class class wbr 3929 cfv 5123 cr 7619 cle 7801 cz 9054 cuz 9326 |
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-pre-ltwlin 7733 |
This theorem depends on definitions: df-bi 116 df-3or 963 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-sbc 2910 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-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-neg 7936 df-z 9055 df-uz 9327 |
This theorem is referenced by: eluz2nn 9364 uzuzle23 9366 eluzge3nn 9367 |
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