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Mirrors > Home > ILE Home > Th. List > nngt0 | GIF version |
Description: A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
Ref | Expression |
---|---|
nngt0 | ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 8855 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
2 | nnge1 8871 | . 2 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
3 | 0lt1 8016 | . . 3 ⊢ 0 < 1 | |
4 | 0re 7890 | . . . 4 ⊢ 0 ∈ ℝ | |
5 | 1re 7889 | . . . 4 ⊢ 1 ∈ ℝ | |
6 | ltletr 7979 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴)) | |
7 | 4, 5, 6 | mp3an12 1316 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴)) |
8 | 3, 7 | mpani 427 | . 2 ⊢ (𝐴 ∈ ℝ → (1 ≤ 𝐴 → 0 < 𝐴)) |
9 | 1, 2, 8 | sylc 62 | 1 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 class class class wbr 3976 ℝcr 7743 0cc0 7744 1c1 7745 < clt 7924 ≤ cle 7925 ℕcn 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-ltwlin 7857 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: nnap0 8877 nngt0i 8878 nn2ge 8881 nn1gt1 8882 nnsub 8887 nngt0d 8892 nnrecl 9103 nn0ge0 9130 0mnnnnn0 9137 elnnnn0b 9149 elnnz 9192 elnn0z 9195 ztri3or0 9224 nnm1ge0 9268 gtndiv 9277 elpq 9577 elpqb 9578 nnrp 9590 nnledivrp 9693 fzo1fzo0n0 10108 ubmelfzo 10125 adddivflid 10217 flltdivnn0lt 10229 intfracq 10245 zmodcl 10269 zmodfz 10271 zmodid2 10277 m1modnnsub1 10295 expnnval 10448 nnlesq 10548 facdiv 10640 faclbnd 10643 bc0k 10658 dvdsval3 11717 nndivdvds 11722 moddvds 11725 evennn2n 11805 nnoddm1d2 11832 divalglemnn 11840 ndvdssub 11852 ndvdsadd 11853 modgcd 11909 sqgcd 11947 lcmgcdlem 11988 qredeu 12008 divdenle 12108 hashgcdlem 12149 oddprm 12170 pythagtriplem12 12186 pythagtriplem13 12187 pythagtriplem14 12188 pythagtriplem16 12190 pythagtriplem19 12193 pc2dvds 12240 fldivp1 12257 znnen 12274 exmidunben 12302 |
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