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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 9207 | . 2 ⊢ ℕ0 = {𝑘 ∈ ℤ ∣ 0 ≤ 𝑘} | |
2 | 0z 9193 | . . 3 ⊢ 0 ∈ ℤ | |
3 | uzval 9459 | . . 3 ⊢ (0 ∈ ℤ → (ℤ≥‘0) = {𝑘 ∈ ℤ ∣ 0 ≤ 𝑘}) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (ℤ≥‘0) = {𝑘 ∈ ℤ ∣ 0 ≤ 𝑘} |
5 | 1, 4 | eqtr4i 2188 | 1 ⊢ ℕ0 = (ℤ≥‘0) |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 {crab 2446 class class class wbr 3976 ‘cfv 5182 0cc0 7744 ≤ cle 7925 ℕ0cn0 9105 ℤcz 9182 ℤ≥cuz 9457 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 |
This theorem is referenced by: elnn0uz 9494 2eluzge0 9504 eluznn0 9528 fseq1p1m1 10019 fz01or 10036 fznn0sub2 10053 nn0split 10061 fzossnn0 10100 frecfzennn 10351 frechashgf1o 10353 exple1 10501 bcval5 10665 bcpasc 10668 hashcl 10683 hashfzo0 10725 zfz1isolemsplit 10737 binom1dif 11414 isumnn0nn 11420 arisum2 11426 expcnvre 11430 explecnv 11432 geoserap 11434 geolim 11438 geolim2 11439 geoisum 11444 geoisumr 11445 mertenslemub 11461 mertenslemi1 11462 mertenslem2 11463 mertensabs 11464 efcllemp 11585 ef0lem 11587 efval 11588 eff 11590 efcvg 11593 efcvgfsum 11594 reefcl 11595 ege2le3 11598 efcj 11600 eftlcvg 11614 eftlub 11617 effsumlt 11619 ef4p 11621 efgt1p2 11622 efgt1p 11623 eflegeo 11628 eirraplem 11703 alginv 11958 algcvg 11959 algcvga 11962 algfx 11963 eucalgcvga 11969 eucalg 11970 phiprmpw 12131 prmdiv 12144 pcfac 12257 ennnfonelemh 12274 ennnfonelemp1 12276 ennnfonelemom 12278 ennnfonelemkh 12282 ennnfonelemrn 12289 dveflem 13228 |
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