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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 8790 | . 2 | |
2 | 2pos 8811 | . 2 | |
3 | 1, 2 | gt0ap0ii 8390 | 1 # |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3929 cc0 7620 # cap 8343 c2 8771 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-2 8779 |
This theorem is referenced by: 2div2e1 8852 4d2e2 8880 halfre 8933 1mhlfehlf 8938 halfpm6th 8940 2muliap0 8944 halfcl 8946 rehalfcl 8947 half0 8948 2halves 8949 halfaddsub 8954 xp1d2m1eqxm1d2 8972 div4p1lem1div2 8973 zneo 9152 nneoor 9153 nneo 9154 zeo 9156 zeo2 9157 halfthird 9324 qbtwnrelemcalc 10033 2tnp1ge0ge0 10074 zesq 10410 sqoddm1div8 10444 faclbnd2 10488 crre 10629 addcj 10663 resqrexlemover 10782 resqrexlemcalc1 10786 resqrexlemcvg 10791 maxabslemab 10978 max0addsup 10991 minabs 11007 bdtri 11011 arisum 11267 arisum2 11268 geo2sum 11283 geo2lim 11285 geoihalfsum 11291 ege2le3 11377 efgt0 11390 tanval2ap 11420 tanval3ap 11421 efi4p 11424 efival 11439 cosadd 11444 sinmul 11451 cosmul 11452 sin01bnd 11464 cos01bnd 11465 sin02gt0 11470 odd2np1 11570 mulsucdiv2z 11582 ltoddhalfle 11590 halfleoddlt 11591 nn0enne 11599 nn0o 11604 flodddiv4 11631 flodddiv4t2lthalf 11634 6lcm4e12 11768 sqrt2irrlem 11839 sqrt2irr 11840 oddennn 11905 evenennn 11906 coscn 12859 sinhalfpilem 12872 cospi 12881 ptolemy 12905 sincosq3sgn 12909 sincosq4sgn 12910 sinq12gt0 12911 cosq23lt0 12914 coseq00topi 12916 tangtx 12919 sincos4thpi 12921 sincos6thpi 12923 sincos3rdpi 12924 pigt3 12925 abssinper 12927 coskpi 12929 |
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