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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 7893 | . . 3 | |
2 | dfnn2 8880 | . . . 4 | |
3 | 2 | rexeqi 2670 | . . 3 |
4 | 1, 3 | sylibr 133 | . 2 |
5 | nnre 8885 | . . . 4 | |
6 | ltxrlt 7985 | . . . 4 | |
7 | 5, 6 | sylan2 284 | . . 3 |
8 | 7 | rexbidva 2467 | . 2 |
9 | 4, 8 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2141 cab 2156 wral 2448 wrex 2449 cint 3831 class class class wbr 3989 (class class class)co 5853 cr 7773 c1 7775 caddc 7777 cltrr 7778 clt 7954 cn 8878 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 ax-arch 7893 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-inn 8879 |
This theorem is referenced by: nnrecl 9133 bndndx 9134 btwnz 9331 expnbnd 10599 cvg1nlemres 10949 cvg1n 10950 resqrexlemga 10987 fsum3cvg3 11359 divcnv 11460 efcllem 11622 alzdvds 11814 dvdsbnd 11911 |
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