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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 7733 | . . 3 ⊢ 1 ∈ ℝ | |
2 | 0lt1 7857 | . . 3 ⊢ 0 < 1 | |
3 | 1, 1, 2, 2 | addgt0ii 8221 | . 2 ⊢ 0 < (1 + 1) |
4 | df-2 8747 | . 2 ⊢ 2 = (1 + 1) | |
5 | 3, 4 | breqtrri 3925 | 1 ⊢ 0 < 2 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3899 (class class class)co 5742 0cc0 7588 1c1 7589 + caddc 7591 < clt 7768 2c2 8739 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-2 8747 |
This theorem is referenced by: 2ne0 8780 2ap0 8781 3pos 8782 halfgt0 8903 halflt1 8905 halfpos2 8918 halfnneg2 8920 nominpos 8925 avglt1 8926 avglt2 8927 nn0n0n1ge2b 9098 3halfnz 9116 2rp 9414 xleaddadd 9638 2tnp1ge0ge0 10042 mulp1mod1 10106 amgm2 10858 cos2bnd 11394 sin02gt0 11397 sincos2sgn 11399 sin4lt0 11400 epos 11414 oexpneg 11501 oddge22np1 11505 evennn02n 11506 nn0ehalf 11527 nno 11530 nn0oddm1d2 11533 nnoddm1d2 11534 flodddiv4t2lthalf 11561 sqrt2re 11768 sqrt2irrap 11785 bl2in 12499 pilem3 12791 pipos 12796 sinhalfpilem 12799 sincosq1lem 12833 sinq12gt0 12838 coseq00topi 12843 coseq0negpitopi 12844 tangtx 12846 sincos4thpi 12848 tan4thpi 12849 sincos6thpi 12850 cosordlem 12857 cos02pilt1 12859 ex-fl 12864 |
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