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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 8631 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
2 | nnge1 8647 | . 2 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
3 | 0lt1 7806 | . . 3 ⊢ 0 < 1 | |
4 | 0re 7684 | . . . 4 ⊢ 0 ∈ ℝ | |
5 | 1re 7683 | . . . 4 ⊢ 1 ∈ ℝ | |
6 | ltletr 7770 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴)) | |
7 | 4, 5, 6 | mp3an12 1286 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴)) |
8 | 3, 7 | mpani 424 | . 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 1461 class class class wbr 3893 ℝcr 7540 0cc0 7541 1c1 7542 < clt 7718 ≤ cle 7719 ℕcn 8624 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1re 7633 ax-addrcl 7636 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-ltadd 7655 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-xp 4503 df-cnv 4505 df-iota 5044 df-fv 5087 df-ov 5729 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-inn 8625 |
This theorem is referenced by: nnap0 8653 nngt0i 8654 nn2ge 8657 nn1gt1 8658 nnsub 8663 nngt0d 8668 nnrecl 8873 nn0ge0 8900 0mnnnnn0 8907 elnnnn0b 8919 elnnz 8962 elnn0z 8965 ztri3or0 8994 nnm1ge0 9035 gtndiv 9044 nnrp 9346 nnledivrp 9440 fzo1fzo0n0 9847 ubmelfzo 9864 adddivflid 9952 flltdivnn0lt 9964 intfracq 9980 zmodcl 10004 zmodfz 10006 zmodid2 10012 m1modnnsub1 10030 expnnval 10183 nnlesq 10283 facdiv 10371 faclbnd 10374 bc0k 10389 dvdsval3 11339 nndivdvds 11341 moddvds 11344 evennn2n 11422 nnoddm1d2 11449 divalglemnn 11457 ndvdssub 11469 ndvdsadd 11470 modgcd 11521 sqgcd 11557 lcmgcdlem 11598 qredeu 11618 divdenle 11714 hashgcdlem 11742 znnen 11750 exmidunben 11778 |
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