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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 8861 | . 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 2128 ≠ wne 2327 0cc0 7732 ℕcn 8833 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1re 7826 ax-addrcl 7829 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-pre-ltirr 7844 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-xp 4592 df-cnv 4594 df-iota 5135 df-fv 5178 df-ov 5827 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-inn 8834 |
This theorem is referenced by: eluz2n0 9481 flqdiv 10220 modsumfzodifsn 10295 facne0 10611 gcdnncl 11851 gcdeq0 11861 dvdsgcdidd 11878 mulgcd 11900 sqgcd 11913 lcmeq0 11948 lcmgcdlem 11954 qredeu 11974 cncongr1 11980 prmind2 11997 divgcdodd 12018 oddpwdclemxy 12044 oddpwdclemodd 12047 divnumden 12071 hashdvds 12096 |
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