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Mirrors > Home > ILE Home > Th. List > arch | Unicode version |
Description: Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.) |
Ref | Expression |
---|---|
arch |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-arch 7003 | . . 3 | |
2 | dfnn2 7916 | . . . 4 | |
3 | 2 | rexeqi 2510 | . . 3 |
4 | 1, 3 | sylibr 137 | . 2 |
5 | nnre 7921 | . . . 4 | |
6 | ltxrlt 7085 | . . . 4 | |
7 | 5, 6 | sylan2 270 | . . 3 |
8 | 7 | rexbidva 2323 | . 2 |
9 | 4, 8 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wcel 1393 cab 2026 wral 2306 wrex 2307 cint 3615 class class class wbr 3764 (class class class)co 5512 cr 6888 c1 6890 caddc 6892 cltrr 6893 clt 7060 cn 7914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1re 6978 ax-addrcl 6981 ax-arch 7003 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-ltxr 7065 df-inn 7915 |
This theorem is referenced by: nnrecl 8179 bndndx 8180 btwnz 8357 expnbnd 9372 cvg1nlemres 9584 cvg1n 9585 resqrexlemga 9621 |
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