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Mirrors > Home > ILE Home > Th. List > nnap0d | Unicode version |
Description: A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
Ref | Expression |
---|---|
nnge1d.1 |
Ref | Expression |
---|---|
nnap0d | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 | |
2 | nnap0 8877 | . 2 # | |
3 | 1, 2 | syl 14 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2135 class class class wbr 3976 cc0 7744 # cap 8470 cn 8848 |
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-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
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-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 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-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-inn 8849 |
This theorem is referenced by: qtri3or 10168 qbtwnrelemcalc 10181 intfracq 10245 flqdiv 10246 modqmulnn 10267 facndiv 10641 bcn0 10657 bcn1 10660 bcm1k 10662 bcp1n 10663 bcp1nk 10664 bcval5 10665 bcpasc 10668 permnn 10673 divcnv 11424 trireciplem 11427 trirecip 11428 expcnvap0 11429 geo2sum 11441 geo2lim 11443 cvgratnnlemfm 11456 cvgratnnlemrate 11457 mertenslemi1 11462 eftabs 11583 efcllemp 11585 ege2le3 11598 efcj 11600 efaddlem 11601 eftlub 11617 eirraplem 11703 dvdsflip 11774 dvdsgcdidd 11912 mulgcd 11934 gcddiv 11937 sqgcd 11947 lcmgcdlem 11988 qredeu 12008 prmind2 12031 isprm5lem 12052 divgcdodd 12054 sqrt2irrlem 12072 oddpwdclemxy 12080 oddpwdclemodd 12083 oddpwdclemdc 12084 sqrt2irraplemnn 12090 sqrt2irrap 12091 qmuldeneqnum 12106 divnumden 12107 numdensq 12113 hashdvds 12132 phiprmpw 12133 pythagtriplem19 12193 pcprendvds2 12202 pcpremul 12204 pceulem 12205 pceu 12206 pcdiv 12213 pcqmul 12214 pcid 12234 pc2dvds 12240 dvdsprmpweqle 12247 pcaddlem 12249 pcadd 12250 oddprmdvds 12263 pockthlem 12265 logbgcd1irraplemexp 13433 logbgcd1irraplemap 13434 cvgcmp2nlemabs 13752 redcwlpolemeq1 13774 |
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