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Mirrors > Home > ILE Home > Th. List > nnne0 | Unicode version |
Description: A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
Ref | Expression |
---|---|
nnne0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nnn 8944 |
. . 3
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2 | eleq1 2240 |
. . 3
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3 | 1, 2 | mtbiri 675 |
. 2
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4 | 3 | necon2ai 2401 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-xp 4632 df-cnv 4634 df-iota 5178 df-fv 5224 df-ov 5877 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-inn 8918 |
This theorem is referenced by: nnne0d 8962 divfnzn 9619 qreccl 9640 fzo1fzo0n0 10180 expnnval 10520 expnegap0 10525 hashnncl 10770 ef0lem 11663 dvdsval3 11793 nndivdvds 11798 modmulconst 11825 dvdsdivcl 11850 divalg2 11925 ndvdssub 11929 nndvdslegcd 11960 divgcdz 11966 divgcdnn 11970 gcdzeq 12017 eucalgf 12049 eucalginv 12050 lcmgcdlem 12071 qredeu 12091 cncongr1 12097 cncongr2 12098 divnumden 12190 divdenle 12191 phimullem 12219 hashgcdlem 12232 phisum 12234 prm23lt5 12257 pythagtriplem8 12266 pythagtriplem9 12267 pceu 12289 pccl 12293 pcdiv 12296 pcqcl 12300 pcdvds 12308 pcndvds 12310 pcndvds2 12312 pceq0 12315 pcz 12325 pcmpt 12335 fldivp1 12340 pcfac 12342 ennnfonelemjn 12397 mulgnn 12943 mulgnegnn 12947 dvexp2 14069 lgsval4a 14316 lgsabs1 14333 lgssq2 14335 |
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