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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 9303 | . 2 ⊢ ℕ0 = {𝑘 ∈ ℤ ∣ 0 ≤ 𝑘} | |
2 | 0z 9289 | . . 3 ⊢ 0 ∈ ℤ | |
3 | uzval 9555 | . . 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 2213 | 1 ⊢ ℕ0 = (ℤ≥‘0) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 {crab 2472 class class class wbr 4018 ‘cfv 5232 0cc0 7836 ≤ cle 8018 ℕ0cn0 9201 ℤcz 9278 ℤ≥cuz 9553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-addass 7938 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-ltadd 7952 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-inn 8945 df-n0 9202 df-z 9279 df-uz 9554 |
This theorem is referenced by: elnn0uz 9590 2eluzge0 9600 eluznn0 9624 fseq1p1m1 10119 fz01or 10136 fznn0sub2 10153 nn0split 10161 fzossnn0 10200 frecfzennn 10452 frechashgf1o 10454 exple1 10602 bcval5 10770 bcpasc 10773 hashcl 10788 hashfzo0 10830 zfz1isolemsplit 10845 binom1dif 11522 isumnn0nn 11528 arisum2 11534 expcnvre 11538 explecnv 11540 geoserap 11542 geolim 11546 geolim2 11547 geoisum 11552 geoisumr 11553 mertenslemub 11569 mertenslemi1 11570 mertenslem2 11571 mertensabs 11572 efcllemp 11693 ef0lem 11695 efval 11696 eff 11698 efcvg 11701 efcvgfsum 11702 reefcl 11703 ege2le3 11706 efcj 11708 eftlcvg 11722 eftlub 11725 effsumlt 11727 ef4p 11729 efgt1p2 11730 efgt1p 11731 eflegeo 11736 eirraplem 11811 alginv 12074 algcvg 12075 algcvga 12078 algfx 12079 eucalgcvga 12085 eucalg 12086 phiprmpw 12249 prmdiv 12262 pcfac 12377 ennnfonelemh 12450 ennnfonelemp1 12452 ennnfonelemom 12454 ennnfonelemkh 12458 ennnfonelemrn 12465 dveflem 14624 lgseisenlem1 14887 |
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