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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 9071 | . 2 ⊢ ℕ = {𝑘 ∈ ℤ ∣ 1 ≤ 𝑘} | |
2 | 1z 9073 | . . 3 ⊢ 1 ∈ ℤ | |
3 | uzval 9321 | . . 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 2161 | 1 ⊢ ℕ = (ℤ≥‘1) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {crab 2418 class class class wbr 3924 ‘cfv 5118 1c1 7614 ≤ cle 7794 ℕcn 8713 ℤcz 9047 ℤ≥cuz 9319 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-z 9048 df-uz 9320 |
This theorem is referenced by: elnnuz 9355 eluz2nn 9357 uznnssnn 9365 eluznn 9387 fzssnn 9841 fseq1p1m1 9867 fz01or 9884 nnsplit 9907 elfzo1 9960 exp3vallem 10287 exp3val 10288 facnn 10466 fac0 10467 bcm1k 10499 bcval5 10502 bcpasc 10505 seq3coll 10578 recvguniq 10760 resqrexlemf 10772 climuni 11055 climrecvg1n 11110 climcvg1nlem 11111 summodclem3 11142 summodclem2a 11143 fsum3 11149 sum0 11150 isumz 11151 fsumcl2lem 11160 fsumadd 11168 fsummulc2 11210 isumnn0nn 11255 divcnv 11259 trireciplem 11262 trirecip 11263 expcnvap0 11264 expcnv 11266 geo2lim 11278 geoisum1 11281 geoisum1c 11282 cvgratnnlemnexp 11286 cvgratnnlemseq 11288 cvgratnnlemrate 11292 cvgratnn 11293 mertenslem2 11298 ege2le3 11366 gcdsupex 11635 gcdsupcl 11636 lcmval 11733 lcmcllem 11737 lcmledvds 11740 isprm3 11788 phicl2 11879 phibndlem 11881 ennnfonelemjn 11904 lmtopcnp 12408 cvgcmp2nlemabs 13216 cvgcmp2n 13217 trilpolemcl 13219 trilpolemisumle 13220 trilpolemgt1 13221 trilpolemeq1 13222 trilpolemlt1 13223 |
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