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Mirrors > Home > ILE Home > Th. List > nnne0d | GIF version |
Description: A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnne0d | ⊢ (𝜑 → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnne0 8945 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ≠ wne 2347 0cc0 7810 ℕcn 8917 |
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: eluz2n0 9568 flqdiv 10318 modsumfzodifsn 10393 facne0 10712 gcdnncl 11962 gcdeq0 11972 dvdsgcdidd 11989 mulgcd 12011 sqgcd 12024 lcmeq0 12065 lcmgcdlem 12071 qredeu 12091 cncongr1 12097 prmind2 12114 isprm5lem 12135 divgcdodd 12137 oddpwdclemxy 12163 oddpwdclemodd 12166 divnumden 12190 hashdvds 12215 pythagtriplem4 12262 pythagtriplem19 12276 pcprendvds2 12285 pcpremul 12287 pceulem 12288 pcqmul 12297 pc2dvds 12323 pcaddlem 12332 pcadd 12333 pcmpt2 12336 pcmptdvds 12337 pcbc 12343 expnprm 12345 prmpwdvds 12347 pockthlem 12348 4sqlem8 12377 4sqlem9 12378 4sqlem10 12379 lgsval2lem 14304 2sqlem3 14346 2sqlem8 14352 |
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