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Mirrors > Home > MPE Home > Th. List > fzssuz | Structured version Visualization version GIF version |
Description: A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.) |
Ref | Expression |
---|---|
fzssuz | ⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz 12894 | . 2 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
2 | 1 | ssriv 3970 | 1 ⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3935 ‘cfv 6349 (class class class)co 7145 ℤ≥cuz 12232 ...cfz 12882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-neg 10862 df-z 11971 df-uz 12233 df-fz 12883 |
This theorem is referenced by: ltwefz 13321 seqcoll2 13813 caubnd 14708 climsup 15016 summolem2a 15062 fsumss 15072 fsumsers 15075 isumclim3 15104 binomlem 15174 prodmolem2a 15278 fprodntriv 15286 fprodss 15292 iprodclim3 15344 fprodefsum 15438 isprm3 16017 2prm 16026 prmreclem5 16246 4sqlem11 16281 gsumval3 18958 telgsums 19044 fz2ssnn0 30435 esumpcvgval 31237 esumcvg 31245 eulerpartlemsv3 31519 ballotlemfc0 31650 ballotlemfcc 31651 ballotlemiex 31659 ballotlemsdom 31669 ballotlemsima 31673 ballotlemrv2 31679 fsum2dsub 31778 erdszelem4 32339 erdszelem8 32343 volsupnfl 34819 sdclem2 34900 geomcau 34917 diophin 39249 irrapxlem1 39299 fzssnn0 41465 iuneqfzuzlem 41482 fzossuz 41530 uzublem 41584 climinf 41767 sge0uzfsumgt 42607 iundjiun 42623 caratheodorylem1 42689 |
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