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Mirrors > Home > ILE Home > Th. List > nnuz | GIF version |
Description: Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
Ref | Expression |
---|---|
nnuz | ⊢ ℕ = (ℤ≥‘1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnzrab 9078 | . 2 ⊢ ℕ = {𝑘 ∈ ℤ ∣ 1 ≤ 𝑘} | |
2 | 1z 9080 | . . 3 ⊢ 1 ∈ ℤ | |
3 | uzval 9328 | . . 3 ⊢ (1 ∈ ℤ → (ℤ≥‘1) = {𝑘 ∈ ℤ ∣ 1 ≤ 𝑘}) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (ℤ≥‘1) = {𝑘 ∈ ℤ ∣ 1 ≤ 𝑘} |
5 | 1, 4 | eqtr4i 2163 | 1 ⊢ ℕ = (ℤ≥‘1) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {crab 2420 class class class wbr 3929 ‘cfv 5123 1c1 7621 ≤ cle 7801 ℕcn 8720 ℤcz 9054 ℤ≥cuz 9326 |
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-mpt 3991 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 df-uz 9327 |
This theorem is referenced by: elnnuz 9362 eluz2nn 9364 uznnssnn 9372 eluznn 9394 fzssnn 9848 fseq1p1m1 9874 fz01or 9891 nnsplit 9914 elfzo1 9967 exp3vallem 10294 exp3val 10295 facnn 10473 fac0 10474 bcm1k 10506 bcval5 10509 bcpasc 10512 seq3coll 10585 recvguniq 10767 resqrexlemf 10779 climuni 11062 climrecvg1n 11117 climcvg1nlem 11118 summodclem3 11149 summodclem2a 11150 fsum3 11156 sum0 11157 isumz 11158 fsumcl2lem 11167 fsumadd 11175 fsummulc2 11217 isumnn0nn 11262 divcnv 11266 trireciplem 11269 trirecip 11270 expcnvap0 11271 expcnv 11273 geo2lim 11285 geoisum1 11288 geoisum1c 11289 cvgratnnlemnexp 11293 cvgratnnlemseq 11295 cvgratnnlemrate 11299 cvgratnn 11300 mertenslem2 11305 prodmodclem3 11344 prodmodclem2a 11345 ege2le3 11377 gcdsupex 11646 gcdsupcl 11647 lcmval 11744 lcmcllem 11748 lcmledvds 11751 isprm3 11799 phicl2 11890 phibndlem 11892 ennnfonelemjn 11915 lmtopcnp 12419 cvgcmp2nlemabs 13227 cvgcmp2n 13228 trilpolemcl 13230 trilpolemisumle 13231 trilpolemgt1 13232 trilpolemeq1 13233 trilpolemlt1 13234 |
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