![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 8924 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
2 | nnge1 8940 | . 2 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
3 | 0lt1 8082 | . . 3 ⊢ 0 < 1 | |
4 | 0re 7956 | . . . 4 ⊢ 0 ∈ ℝ | |
5 | 1re 7955 | . . . 4 ⊢ 1 ∈ ℝ | |
6 | ltletr 8045 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴)) | |
7 | 4, 5, 6 | mp3an12 1327 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝐴) → 0 < 𝐴)) |
8 | 3, 7 | mpani 430 | . 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 104 ∈ wcel 2148 class class class wbr 4003 ℝcr 7809 0cc0 7810 1c1 7811 < clt 7990 ≤ cle 7991 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-ltwlin 7923 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: nnap0 8946 nngt0i 8947 nn2ge 8950 nn1gt1 8951 nnsub 8956 nngt0d 8961 nnrecl 9172 nn0ge0 9199 0mnnnnn0 9206 elnnnn0b 9218 elnnz 9261 elnn0z 9264 ztri3or0 9293 nnm1ge0 9337 gtndiv 9346 elpq 9646 elpqb 9647 nnrp 9661 nnledivrp 9764 fzo1fzo0n0 10180 ubmelfzo 10197 adddivflid 10289 flltdivnn0lt 10301 intfracq 10317 zmodcl 10341 zmodfz 10343 zmodid2 10349 m1modnnsub1 10367 expnnval 10520 nnlesq 10620 facdiv 10713 faclbnd 10716 bc0k 10731 dvdsval3 11793 nndivdvds 11798 moddvds 11801 evennn2n 11882 nnoddm1d2 11909 divalglemnn 11917 ndvdssub 11929 ndvdsadd 11930 modgcd 11986 sqgcd 12024 lcmgcdlem 12071 qredeu 12091 divdenle 12191 hashgcdlem 12232 oddprm 12253 pythagtriplem12 12269 pythagtriplem13 12270 pythagtriplem14 12271 pythagtriplem16 12273 pythagtriplem19 12276 pc2dvds 12323 fldivp1 12340 znnen 12393 exmidunben 12421 mulgnn 12943 mulgnegnn 12947 mulgmodid 12975 lgsval4a 14316 lgsne0 14332 |
Copyright terms: Public domain | W3C validator |