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Mirrors > Home > ILE Home > Th. List > 0lt1 | Unicode version |
Description: 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.) |
Ref | Expression |
---|---|
0lt1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-0lt1 6990 | . 2 | |
2 | 0re 7027 | . . 3 | |
3 | 1re 7026 | . . 3 | |
4 | ltxrlt 7085 | . . 3 | |
5 | 2, 3, 4 | mp2an 402 | . 2 |
6 | 1, 5 | mpbir 134 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 98 wcel 1393 class class class wbr 3764 cr 6888 cc0 6889 c1 6890 cltrr 6893 clt 7060 |
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-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1re 6978 ax-addrcl 6981 ax-rnegex 6993 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 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-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-ltxr 7065 |
This theorem is referenced by: ine0 7391 0le1 7476 inelr 7575 1ap0 7581 eqneg 7708 ltp1 7810 ltm1 7812 recgt0 7816 mulgt1 7829 reclt1 7862 recgt1 7863 recgt1i 7864 recp1lt1 7865 recreclt 7866 nnge1 7937 nngt0 7939 0nnn 7941 nnrecgt0 7951 0ne1 7982 2pos 8007 3pos 8010 4pos 8013 5pos 8016 6pos 8017 7pos 8018 8pos 8019 9pos 8020 10pos 8021 neg1lt0 8025 halflt1 8142 nn0p1gt0 8211 elnnnn0c 8227 elnnz1 8268 recnz 8333 1rp 8587 divlt1lt 8650 divle1le 8651 ledivge1le 8652 fz10 8910 fzpreddisj 8933 elfz1b 8952 modqfrac 9179 expgt1 9293 ltexp2a 9306 leexp2a 9307 expnbnd 9372 expnlbnd 9373 expnlbnd2 9374 resqrexlem1arp 9603 mulcn2 9833 |
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