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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 8906 | . 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 2141 ≠ wne 2340 0cc0 7774 ℕcn 8878 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-pre-ltirr 7886 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-iota 5160 df-fv 5206 df-ov 5856 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-inn 8879 |
This theorem is referenced by: eluz2n0 9529 flqdiv 10277 modsumfzodifsn 10352 facne0 10671 gcdnncl 11922 gcdeq0 11932 dvdsgcdidd 11949 mulgcd 11971 sqgcd 11984 lcmeq0 12025 lcmgcdlem 12031 qredeu 12051 cncongr1 12057 prmind2 12074 isprm5lem 12095 divgcdodd 12097 oddpwdclemxy 12123 oddpwdclemodd 12126 divnumden 12150 hashdvds 12175 pythagtriplem4 12222 pythagtriplem19 12236 pcprendvds2 12245 pcpremul 12247 pceulem 12248 pcqmul 12257 pc2dvds 12283 pcaddlem 12292 pcadd 12293 pcmpt2 12296 pcmptdvds 12297 pcbc 12303 expnprm 12305 prmpwdvds 12307 pockthlem 12308 4sqlem8 12337 4sqlem9 12338 4sqlem10 12339 lgsval2lem 13705 2sqlem3 13747 2sqlem8 13753 |
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