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Mirrors > Home > ILE Home > Th. List > nnz | GIF version |
Description: A positive integer is an integer. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nnz | ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssz 9071 | . 2 ⊢ ℕ ⊆ ℤ | |
2 | 1 | sseli 3093 | 1 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ℕcn 8720 ℤcz 9054 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-z 9055 |
This theorem is referenced by: elnnz1 9077 znegcl 9085 nnleltp1 9113 nnltp1le 9114 elz2 9122 nnlem1lt 9135 nnltlem1 9136 nnm1ge0 9137 prime 9150 nneo 9154 zeo 9156 btwnz 9170 indstr 9388 eluz2b2 9397 elnn1uz2 9401 qaddcl 9427 qreccl 9434 elfz1end 9835 fznatpl1 9856 fznn 9869 elfz1b 9870 elfzo0 9959 fzo1fzo0n0 9960 elfzo0z 9961 elfzo1 9967 ubmelm1fzo 10003 intfracq 10093 zmodcl 10117 zmodfz 10119 zmodfzo 10120 zmodid2 10125 zmodidfzo 10126 modfzo0difsn 10168 mulexpzap 10333 nnesq 10411 expnlbnd 10416 expnlbnd2 10417 facdiv 10484 faclbnd 10487 bc0k 10502 bcval5 10509 seq3coll 10585 caucvgrelemcau 10752 resqrexlemlo 10785 resqrexlemcalc3 10788 resqrexlemgt0 10792 absexpzap 10852 climuni 11062 fsum3 11156 arisum 11267 trireciplem 11269 expcnvap0 11271 geo2sum 11283 geo2lim 11285 0.999... 11290 geoihalfsum 11291 cvgratz 11301 dvdsval3 11497 nndivdvds 11499 modmulconst 11525 dvdsle 11542 dvdsssfz1 11550 fzm1ndvds 11554 dvdsfac 11558 oexpneg 11574 nnoddm1d2 11607 divalg2 11623 divalgmod 11624 modremain 11626 ndvdsadd 11628 nndvdslegcd 11654 divgcdz 11660 divgcdnn 11663 divgcdnnr 11664 modgcd 11679 gcddiv 11707 gcdmultiple 11708 gcdmultiplez 11709 gcdzeq 11710 gcdeq 11711 rpmulgcd 11714 rplpwr 11715 rppwr 11716 sqgcd 11717 dvdssqlem 11718 dvdssq 11719 eucalginv 11737 lcmgcdlem 11758 lcmgcdnn 11763 lcmass 11766 coprmgcdb 11769 qredeq 11777 qredeu 11778 cncongr1 11784 cncongr2 11785 1idssfct 11796 isprm2lem 11797 isprm3 11799 isprm4 11800 prmind2 11801 divgcdodd 11821 isprm6 11825 sqrt2irr 11840 pw2dvds 11844 sqrt2irraplemnn 11857 divnumden 11874 divdenle 11875 nn0gcdsq 11878 phivalfi 11888 phicl2 11890 phiprmpw 11898 hashgcdlem 11903 hashgcdeq 11904 oddennn 11905 evenennn 11906 unennn 11910 trilpolemcl 13230 |
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