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Mirrors > Home > ILE Home > Th. List > nngt0d | GIF version |
Description: A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nngt0d | ⊢ (𝜑 → 0 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nngt0 8873 | . 2 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 0 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 class class class wbr 3976 0cc0 7744 < clt 7924 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: flqdiv 10246 modqmulnn 10267 modifeq2int 10311 modaddmodup 10312 modaddmodlo 10313 modsumfzodifsn 10321 addmodlteq 10323 facubnd 10647 resqrexlemdecn 10940 modfsummodlemstep 11384 divcnv 11424 cvgratnnlemabsle 11454 fprodmodd 11568 efcllemp 11585 ege2le3 11598 eftlub 11617 eflegeo 11628 eirraplem 11703 dvdslelemd 11766 dvdsmod 11785 mulmoddvds 11786 divalgmod 11849 bezoutlemnewy 11914 bezoutlemstep 11915 sqgcd 11947 eucalglt 11968 qredeu 12008 prmind2 12031 nprm 12034 sqrt2irraplemnn 12090 divdenle 12108 qnumgt0 12109 hashdvds 12132 crth 12135 phimullem 12136 eulerthlema 12141 fermltl 12145 prmdiv 12146 prmdiveq 12147 odzdvds 12156 powm2modprm 12163 modprm0 12165 nnnn0modprm0 12166 pythagtriplem11 12185 pythagtriplem13 12187 pythagtriplem19 12193 pcadd 12250 pcfaclem 12258 qexpz 12261 pockthlem 12265 pockthg 12266 |
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