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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 7851 | . . 3 | |
2 | dfnn2 8835 | . . . 4 | |
3 | 2 | rexeqi 2657 | . . 3 |
4 | 1, 3 | sylibr 133 | . 2 |
5 | nnre 8840 | . . . 4 | |
6 | ltxrlt 7943 | . . . 4 | |
7 | 5, 6 | sylan2 284 | . . 3 |
8 | 7 | rexbidva 2454 | . 2 |
9 | 4, 8 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2128 cab 2143 wral 2435 wrex 2436 cint 3807 class class class wbr 3965 (class class class)co 5824 cr 7731 c1 7733 caddc 7735 cltrr 7736 clt 7912 cn 8833 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1re 7826 ax-addrcl 7829 ax-arch 7851 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-xp 4592 df-pnf 7914 df-mnf 7915 df-ltxr 7917 df-inn 8834 |
This theorem is referenced by: nnrecl 9088 bndndx 9089 btwnz 9283 expnbnd 10541 cvg1nlemres 10885 cvg1n 10886 resqrexlemga 10923 fsum3cvg3 11293 divcnv 11394 efcllem 11556 alzdvds 11745 dvdsbnd 11839 |
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