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Mirrors > Home > ILE Home > Th. List > elfz2nn0 | Unicode version |
Description: Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfz2nn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0uz 9363 | . . . 4 | |
2 | 1 | anbi1i 453 | . . 3 |
3 | eluznn0 9393 | . . . . . 6 | |
4 | eluzle 9338 | . . . . . . 7 | |
5 | 4 | adantl 275 | . . . . . 6 |
6 | 3, 5 | jca 304 | . . . . 5 |
7 | nn0z 9074 | . . . . . . . 8 | |
8 | nn0z 9074 | . . . . . . . 8 | |
9 | eluz 9339 | . . . . . . . 8 | |
10 | 7, 8, 9 | syl2an 287 | . . . . . . 7 |
11 | 10 | biimprd 157 | . . . . . 6 |
12 | 11 | impr 376 | . . . . 5 |
13 | 6, 12 | impbida 585 | . . . 4 |
14 | 13 | pm5.32i 449 | . . 3 |
15 | 2, 14 | bitr3i 185 | . 2 |
16 | elfzuzb 9800 | . 2 | |
17 | 3anass 966 | . 2 | |
18 | 15, 16, 17 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3a 962 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc0 7620 cle 7801 cn0 8977 cz 9054 cuz 9326 cfz 9790 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: elfznn0 9894 elfz3nn0 9895 0elfz 9898 elfz0ubfz0 9902 elfz0fzfz0 9903 fz0fzelfz0 9904 uzsubfz0 9906 fz0fzdiffz0 9907 elfzmlbm 9908 elfzmlbp 9909 difelfzle 9911 difelfznle 9912 fzofzim 9965 elfzodifsumelfzo 9978 elfzom1elp1fzo 9979 fzo0to42pr 9997 fzo0sn0fzo1 9998 fvinim0ffz 10018 1elfz0hash 10552 |
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