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Mirrors > Home > ILE Home > Th. List > 2ap0 | Unicode version |
Description: The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Ref | Expression |
---|---|
2ap0 | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8758 | . 2 | |
2 | 2pos 8779 | . 2 | |
3 | 1, 2 | gt0ap0ii 8358 | 1 # |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3899 cc0 7588 # cap 8311 c2 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-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
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-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 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-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-2 8747 |
This theorem is referenced by: 2div2e1 8820 4d2e2 8848 halfre 8901 1mhlfehlf 8906 halfpm6th 8908 2muliap0 8912 halfcl 8914 rehalfcl 8915 half0 8916 2halves 8917 halfaddsub 8922 xp1d2m1eqxm1d2 8940 div4p1lem1div2 8941 zneo 9120 nneoor 9121 nneo 9122 zeo 9124 zeo2 9125 halfthird 9292 qbtwnrelemcalc 10001 2tnp1ge0ge0 10042 zesq 10378 sqoddm1div8 10412 faclbnd2 10456 crre 10597 addcj 10631 resqrexlemover 10750 resqrexlemcalc1 10754 resqrexlemcvg 10759 maxabslemab 10946 max0addsup 10959 minabs 10975 bdtri 10979 arisum 11235 arisum2 11236 geo2sum 11251 geo2lim 11253 geoihalfsum 11259 ege2le3 11304 efgt0 11317 tanval2ap 11347 tanval3ap 11348 efi4p 11351 efival 11366 cosadd 11371 sinmul 11378 cosmul 11379 sin01bnd 11391 cos01bnd 11392 sin02gt0 11397 odd2np1 11497 mulsucdiv2z 11509 ltoddhalfle 11517 halfleoddlt 11518 nn0enne 11526 nn0o 11531 flodddiv4 11558 flodddiv4t2lthalf 11561 6lcm4e12 11695 sqrt2irrlem 11766 sqrt2irr 11767 oddennn 11832 evenennn 11833 coscn 12786 sinhalfpilem 12799 cospi 12808 ptolemy 12832 sincosq3sgn 12836 sincosq4sgn 12837 sinq12gt0 12838 cosq23lt0 12841 coseq00topi 12843 tangtx 12846 sincos4thpi 12848 sincos6thpi 12850 sincos3rdpi 12851 pigt3 12852 abssinper 12854 coskpi 12856 |
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