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Mirrors > Home > ILE Home > Th. List > 2pos | GIF version |
Description: The number 2 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
2pos | ⊢ 0 < 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7889 | . . 3 ⊢ 1 ∈ ℝ | |
2 | 0lt1 8016 | . . 3 ⊢ 0 < 1 | |
3 | 1, 1, 2, 2 | addgt0ii 8380 | . 2 ⊢ 0 < (1 + 1) |
4 | df-2 8907 | . 2 ⊢ 2 = (1 + 1) | |
5 | 3, 4 | breqtrri 4003 | 1 ⊢ 0 < 2 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3976 (class class class)co 5836 0cc0 7744 1c1 7745 + caddc 7747 < clt 7924 2c2 8899 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-iota 5147 df-fv 5190 df-ov 5839 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-2 8907 |
This theorem is referenced by: 2ne0 8940 2ap0 8941 3pos 8942 halfgt0 9063 halflt1 9065 halfpos2 9078 halfnneg2 9080 nominpos 9085 avglt1 9086 avglt2 9087 nn0n0n1ge2b 9261 3halfnz 9279 2rp 9585 xleaddadd 9814 2tnp1ge0ge0 10226 mulp1mod1 10290 amgm2 11046 cos2bnd 11687 sin02gt0 11690 sincos2sgn 11692 sin4lt0 11693 epos 11707 oexpneg 11799 oddge22np1 11803 evennn02n 11804 nn0ehalf 11825 nno 11828 nn0oddm1d2 11831 nnoddm1d2 11832 flodddiv4t2lthalf 11859 sqrt2re 12074 sqrt2irrap 12091 bl2in 12950 pilem3 13251 pipos 13256 sinhalfpilem 13259 sincosq1lem 13293 sinq12gt0 13298 coseq00topi 13303 coseq0negpitopi 13304 tangtx 13306 sincos4thpi 13308 tan4thpi 13309 sincos6thpi 13310 cosordlem 13317 cos02pilt1 13319 ex-fl 13449 |
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