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Mirrors > Home > ILE Home > Th. List > 0lt1 | GIF 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 | ⊢ 0 < 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-0lt1 7908 | . 2 ⊢ 0 <ℝ 1 | |
2 | 0re 7948 | . . 3 ⊢ 0 ∈ ℝ | |
3 | 1re 7947 | . . 3 ⊢ 1 ∈ ℝ | |
4 | ltxrlt 8013 | . . 3 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 1 ↔ 0 <ℝ 1)) | |
5 | 2, 3, 4 | mp2an 426 | . 2 ⊢ (0 < 1 ↔ 0 <ℝ 1) |
6 | 1, 5 | mpbir 146 | 1 ⊢ 0 < 1 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 class class class wbr 4000 ℝcr 7801 0cc0 7802 1c1 7803 <ℝ cltrr 7806 < clt 7982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 ax-0lt1 7908 ax-rnegex 7911 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4629 df-pnf 7984 df-mnf 7985 df-ltxr 7987 |
This theorem is referenced by: ine0 8341 0le1 8428 inelr 8531 1ap0 8537 eqneg 8678 ltp1 8790 ltm1 8792 recgt0 8796 mulgt1 8809 reclt1 8842 recgt1 8843 recgt1i 8844 recp1lt1 8845 recreclt 8846 sup3exmid 8903 nnge1 8931 nngt0 8933 0nnn 8935 nnrecgt0 8946 0ne1 8975 2pos 8999 3pos 9002 4pos 9005 5pos 9008 6pos 9009 7pos 9010 8pos 9011 9pos 9012 neg1lt0 9016 halflt1 9125 nn0p1gt0 9194 elnnnn0c 9210 elnnz1 9265 recnz 9335 1rp 9644 divlt1lt 9711 divle1le 9712 ledivge1le 9713 nnledivrp 9753 fz10 10032 fzpreddisj 10057 elfz1b 10076 modqfrac 10323 expgt1 10544 ltexp2a 10558 leexp2a 10559 expnbnd 10629 expnlbnd 10630 expnlbnd2 10631 nn0ltexp2 10674 expcanlem 10679 expcan 10680 bcn1 10722 resqrexlem1arp 10998 mulcn2 11304 reccn2ap 11305 georeclim 11505 geoisumr 11510 cos1bnd 11751 sin01gt0 11753 sincos1sgn 11756 p1modz1 11785 nnoddm1d2 11898 dvdsnprmd 12108 divdenle 12180 mopnex 13672 reeff1olem 13859 cos02pilt1 13939 rplogcl 13967 cxplt 14003 cxple 14004 ltexp2 14027 apdiff 14452 |
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