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Mirrors > Home > ILE Home > Th. List > nn0re | GIF version |
Description: A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nn0re | ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssre 8981 | . 2 ⊢ ℕ0 ⊆ ℝ | |
2 | 1 | sseli 3093 | 1 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ℝcr 7619 ℕ0cn0 8977 |
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-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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-int 3772 df-inn 8721 df-n0 8978 |
This theorem is referenced by: nn0nlt0 9003 nn0le0eq0 9005 nn0p1gt0 9006 elnnnn0c 9022 nn0addge1 9023 nn0addge2 9024 nn0ge2m1nn 9037 nn0nndivcl 9039 xnn0xr 9045 nn0nepnf 9048 xnn0nemnf 9051 elnn0z 9067 elznn0nn 9068 nn0lt10b 9131 nn0ge0div 9138 xnn0lenn0nn0 9648 xnn0xadd0 9650 nn0fz0 9899 elfz0fzfz0 9903 fz0fzelfz0 9904 fz0fzdiffz0 9907 fzctr 9910 difelfzle 9911 difelfznle 9912 elfzo0le 9962 fzonmapblen 9964 fzofzim 9965 elfzodifsumelfzo 9978 fzonn0p1 9988 fzonn0p1p1 9990 elfzom1p1elfzo 9991 ubmelm1fzo 10003 fvinim0ffz 10018 subfzo0 10019 adddivflid 10065 divfl0 10069 flltdivnn0lt 10077 addmodid 10145 modfzo0difsn 10168 inftonninf 10214 bernneq 10412 bernneq3 10414 facwordi 10486 faclbnd 10487 faclbnd3 10489 faclbnd6 10490 facubnd 10491 facavg 10492 bcval4 10498 bcval5 10509 bcpasc 10512 fihashneq0 10541 dvdseq 11546 oddge22np1 11578 nn0ehalf 11600 nn0o 11604 nn0oddm1d2 11606 gcdn0gt0 11666 nn0gcdid0 11669 absmulgcd 11705 nn0seqcvgd 11722 algcvgblem 11730 algcvga 11732 lcmgcdnn 11763 prmfac1 11830 nonsq 11885 hashgcdlem 11903 |
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