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Mirrors > Home > ILE Home > Th. List > nn0z | GIF version |
Description: A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nn0z | ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssz 9185 | . 2 ⊢ ℕ0 ⊆ ℤ | |
2 | 1 | sseli 3124 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 ℕ0cn0 9090 ℤcz 9167 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 |
This theorem is referenced by: nn0negz 9201 nn0ltp1le 9229 nn0leltp1 9230 nn0ltlem1 9231 nn0sub 9233 nn0n0n1ge2b 9243 nn0lt10b 9244 nn0lt2 9245 nn0le2is012 9246 nn0lem1lt 9247 fnn0ind 9280 nn0pzuz 9498 nn01to3 9526 nn0ge2m1nnALT 9527 fz1n 9946 ige2m1fz 10012 elfz2nn0 10014 fznn0 10015 elfz0add 10022 fzctr 10032 difelfzle 10033 fzo1fzo0n0 10082 fzofzim 10087 elfzodifsumelfzo 10100 zpnn0elfzo 10106 fzossfzop1 10111 ubmelm1fzo 10125 adddivflid 10191 fldivnn0 10194 divfl0 10195 flqmulnn0 10198 fldivnn0le 10202 zmodidfzoimp 10253 modqmuladdnn0 10267 modifeq2int 10285 modfzo0difsn 10294 uzennn 10335 expdivap 10470 faclbnd3 10617 bccmpl 10628 bcnp1n 10633 bcn2 10638 bcp1m1 10639 modfsummodlemstep 11354 bcxmas 11386 geo2sum2 11412 mertenslemi1 11432 mertensabs 11434 esum 11559 efcvgfsum 11564 ege2le3 11568 eftlcl 11585 reeftlcl 11586 eftlub 11587 effsumlt 11589 eirraplem 11673 dvds1 11744 dvdsext 11746 addmodlteqALT 11750 oddnn02np1 11770 oddge22np1 11771 nn0ehalf 11793 nn0o1gt2 11795 nno 11796 nn0o 11797 nn0oddm1d2 11799 modremain 11819 gcdn0gt0 11861 nn0gcdid0 11864 bezoutlemmain 11881 nn0seqcvgd 11917 algcvgblem 11925 algcvga 11927 eucalgf 11931 prmndvdsfaclt 12030 nn0sqrtelqelz 12080 nonsq 12081 crth 12098 odzdvds 12119 |
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