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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nnn 8875 | . . 3 | |
2 | eleq1 2227 | . . 3 | |
3 | 1, 2 | mtbiri 665 | . 2 |
4 | 3 | necon2ai 2388 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1342 wcel 2135 wne 2334 cc0 7744 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-1re 7838 ax-addrcl 7841 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
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-rab 2451 df-v 2723 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-xp 4604 df-cnv 4606 df-iota 5147 df-fv 5190 df-ov 5839 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-inn 8849 |
This theorem is referenced by: nnne0d 8893 divfnzn 9550 qreccl 9571 fzo1fzo0n0 10108 expnnval 10448 expnegap0 10453 hashnncl 10698 ef0lem 11587 dvdsval3 11717 nndivdvds 11722 modmulconst 11749 dvdsdivcl 11773 divalg2 11848 ndvdssub 11852 nndvdslegcd 11883 divgcdz 11889 divgcdnn 11893 gcdzeq 11940 eucalgf 11966 eucalginv 11967 lcmgcdlem 11988 qredeu 12008 cncongr1 12014 cncongr2 12015 divnumden 12107 divdenle 12108 phimullem 12136 hashgcdlem 12149 phisum 12151 prm23lt5 12174 pythagtriplem8 12183 pythagtriplem9 12184 pceu 12206 pccl 12210 pcdiv 12213 pcqcl 12217 pcdvds 12225 pcndvds 12227 pcndvds2 12229 pceq0 12232 pcz 12242 pcmpt 12252 fldivp1 12257 pcfac 12259 ennnfonelemjn 12278 dvexp2 13223 |
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