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Mirrors > Home > ILE Home > Th. List > nn0uz | GIF version |
Description: Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
Ref | Expression |
---|---|
nn0uz | ⊢ ℕ0 = (ℤ≥‘0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0zrab 8977 | . 2 ⊢ ℕ0 = {𝑘 ∈ ℤ ∣ 0 ≤ 𝑘} | |
2 | 0z 8963 | . . 3 ⊢ 0 ∈ ℤ | |
3 | uzval 9224 | . . 3 ⊢ (0 ∈ ℤ → (ℤ≥‘0) = {𝑘 ∈ ℤ ∣ 0 ≤ 𝑘}) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ (ℤ≥‘0) = {𝑘 ∈ ℤ ∣ 0 ≤ 𝑘} |
5 | 1, 4 | eqtr4i 2136 | 1 ⊢ ℕ0 = (ℤ≥‘0) |
Colors of variables: wff set class |
Syntax hints: = wceq 1312 ∈ wcel 1461 {crab 2392 class class class wbr 3893 ‘cfv 5079 0cc0 7541 ≤ cle 7719 ℕ0cn0 8875 ℤcz 8952 ℤ≥cuz 9222 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-ltadd 7655 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-inn 8625 df-n0 8876 df-z 8953 df-uz 9223 |
This theorem is referenced by: elnn0uz 9259 2eluzge0 9266 eluznn0 9289 fseq1p1m1 9761 fz01or 9778 fznn0sub2 9792 nn0split 9800 fzossnn0 9839 frecfzennn 10086 frechashgf1o 10088 exple1 10236 bcval5 10396 bcpasc 10399 hashcl 10414 hashfzo0 10456 zfz1isolemsplit 10468 binom1dif 11142 isumnn0nn 11148 arisum2 11154 expcnvre 11158 explecnv 11160 geoserap 11162 geolim 11166 geolim2 11167 geoisum 11172 geoisumr 11173 mertenslemub 11189 mertenslemi1 11190 mertenslem2 11191 mertensabs 11192 efcllemp 11209 ef0lem 11211 efval 11212 eff 11214 efcvg 11217 efcvgfsum 11218 reefcl 11219 ege2le3 11222 efcj 11224 eftlcvg 11238 eftlub 11241 effsumlt 11243 ef4p 11245 efgt1p2 11246 efgt1p 11247 eflegeo 11253 eirraplem 11325 alginv 11568 algcvg 11569 algcvga 11572 algfx 11573 eucalgcvga 11579 eucalg 11580 phiprmpw 11737 ennnfonelemh 11756 ennnfonelemp1 11758 ennnfonelemom 11760 ennnfonelemkh 11764 ennnfonelemrn 11771 |
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