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Mirrors > Home > ILE Home > Th. List > nnnn0 | GIF version |
Description: A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nnnn0 | ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssnn0 8948 | . 2 ⊢ ℕ ⊆ ℕ0 | |
2 | 1 | sseli 3063 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ℕcn 8688 ℕ0cn0 8945 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-n0 8946 |
This theorem is referenced by: nnnn0i 8953 elnnnn0b 8989 elnnnn0c 8990 elnn0z 9035 elz2 9090 nn0ind-raph 9136 zindd 9137 fzo1fzo0n0 9928 ubmelfzo 9945 elfzom1elp1fzo 9947 fzo0sn0fzo1 9966 modqmulnn 10083 expnegap0 10269 expcllem 10272 expcl2lemap 10273 expap0 10291 expeq0 10292 mulexpzap 10301 expnlbnd 10384 facdiv 10452 faclbnd 10455 faclbnd3 10457 faclbnd6 10458 resqrexlemlo 10753 absexpzap 10820 isummolemnm 11116 summodclem2a 11118 fsum3 11124 arisum 11235 expcnvap0 11239 expcnv 11241 geo2sum 11251 geo2lim 11253 geoisum1c 11257 0.999... 11258 mertenslem2 11273 ef0lem 11293 ege2le3 11304 efaddlem 11307 efexp 11315 nn0enne 11526 nnehalf 11528 nno 11530 nn0o 11531 divalg2 11550 ndvdssub 11554 gcddiv 11634 gcdmultiple 11635 gcdmultiplez 11636 rpmulgcd 11641 rplpwr 11642 dvdssqlem 11645 eucalgf 11663 1nprm 11722 isprm6 11752 prmdvdsexp 11753 pw2dvds 11771 oddpwdc 11779 phicl2 11817 phibndlem 11819 phiprmpw 11825 crth 11827 hashgcdlem 11830 ennnfonelemjn 11842 dvexp 12771 |
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