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Mirrors > Home > ILE Home > Th. List > nn0uz | Unicode version |
Description: Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
Ref | Expression |
---|---|
nn0uz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0zrab 9237 | . 2 | |
2 | 0z 9223 | . . 3 | |
3 | uzval 9489 | . . 3 | |
4 | 2, 3 | ax-mp 5 | . 2 |
5 | 1, 4 | eqtr4i 2194 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1348 wcel 2141 crab 2452 class class class wbr 3989 cfv 5198 cc0 7774 cle 7955 cn0 9135 cz 9212 cuz 9487 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 |
This theorem is referenced by: elnn0uz 9524 2eluzge0 9534 eluznn0 9558 fseq1p1m1 10050 fz01or 10067 fznn0sub2 10084 nn0split 10092 fzossnn0 10131 frecfzennn 10382 frechashgf1o 10384 exple1 10532 bcval5 10697 bcpasc 10700 hashcl 10715 hashfzo0 10758 zfz1isolemsplit 10773 binom1dif 11450 isumnn0nn 11456 arisum2 11462 expcnvre 11466 explecnv 11468 geoserap 11470 geolim 11474 geolim2 11475 geoisum 11480 geoisumr 11481 mertenslemub 11497 mertenslemi1 11498 mertenslem2 11499 mertensabs 11500 efcllemp 11621 ef0lem 11623 efval 11624 eff 11626 efcvg 11629 efcvgfsum 11630 reefcl 11631 ege2le3 11634 efcj 11636 eftlcvg 11650 eftlub 11653 effsumlt 11655 ef4p 11657 efgt1p2 11658 efgt1p 11659 eflegeo 11664 eirraplem 11739 alginv 12001 algcvg 12002 algcvga 12005 algfx 12006 eucalgcvga 12012 eucalg 12013 phiprmpw 12176 prmdiv 12189 pcfac 12302 ennnfonelemh 12359 ennnfonelemp1 12361 ennnfonelemom 12363 ennnfonelemkh 12367 ennnfonelemrn 12374 dveflem 13481 |
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