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Mirrors > Home > ILE Home > Th. List > ige2m1fz | Unicode version |
Description: Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.) |
Ref | Expression |
---|---|
ige2m1fz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1eluzge0 9526 | . . 3 | |
2 | fzss1 10012 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | 2z 9233 | . . . . 5 | |
5 | 4 | a1i 9 | . . . 4 |
6 | nn0z 9225 | . . . . 5 | |
7 | 6 | adantr 274 | . . . 4 |
8 | simpr 109 | . . . 4 | |
9 | eluz2 9486 | . . . 4 | |
10 | 5, 7, 8, 9 | syl3anbrc 1176 | . . 3 |
11 | ige2m1fz1 10058 | . . 3 | |
12 | 10, 11 | syl 14 | . 2 |
13 | 3, 12 | sselid 3145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2141 wss 3121 class class class wbr 3987 cfv 5196 (class class class)co 5851 cc0 7767 c1 7768 cle 7948 cmin 8083 c2 8922 cn0 9128 cz 9205 cuz 9480 cfz 9958 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-2 8930 df-n0 9129 df-z 9206 df-uz 9481 df-fz 9959 |
This theorem is referenced by: (None) |
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