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Mirrors > Home > ILE Home > Th. List > fzossnn | GIF version |
Description: Half-open integer ranges starting with 1 are subsets of ℕ. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
Ref | Expression |
---|---|
fzossnn | ⊢ (1..^𝑁) ⊆ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzossfz 10138 | . 2 ⊢ (1..^𝑁) ⊆ (1...𝑁) | |
2 | elfznn 10027 | . . 3 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
3 | 2 | ssriv 3159 | . 2 ⊢ (1...𝑁) ⊆ ℕ |
4 | 1, 3 | sstri 3164 | 1 ⊢ (1..^𝑁) ⊆ ℕ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3129 (class class class)co 5868 1c1 7790 ℕcn 8895 ...cfz 9982 ..^cfzo 10115 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-n0 9153 df-z 9230 df-uz 9505 df-fz 9983 df-fzo 10116 |
This theorem is referenced by: elfzo1 10163 |
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