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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 7095 | . . 3 | |
2 | dfnn2 8041 | . . . 4 | |
3 | 2 | rexeqi 2554 | . . 3 |
4 | 1, 3 | sylibr 132 | . 2 |
5 | nnre 8046 | . . . 4 | |
6 | ltxrlt 7178 | . . . 4 | |
7 | 5, 6 | sylan2 280 | . . 3 |
8 | 7 | rexbidva 2365 | . 2 |
9 | 4, 8 | mpbird 165 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wcel 1433 cab 2067 wral 2348 wrex 2349 cint 3636 class class class wbr 3785 (class class class)co 5532 cr 6980 c1 6982 caddc 6984 cltrr 6985 clt 7153 cn 8039 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1re 7070 ax-addrcl 7073 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-inn 8040 |
This theorem is referenced by: nnrecl 8286 bndndx 8287 btwnz 8466 expnbnd 9596 cvg1nlemres 9871 cvg1n 9872 resqrexlemga 9909 alzdvds 10254 dvdsbnd 10348 |
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