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Mirrors > Home > MPE Home > Th. List > fzssz | Structured version Visualization version GIF version |
Description: A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fzssz | ⊢ (𝑀...𝑁) ⊆ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 12911 | . 2 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
2 | 1 | ssriv 3973 | 1 ⊢ (𝑀...𝑁) ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3938 (class class class)co 7158 ℤcz 11984 ...cfz 12895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-neg 10875 df-z 11985 df-uz 12247 df-fz 12896 |
This theorem is referenced by: fzof 13038 fzossz 13060 seqcoll 13825 prodmolem2a 15290 lcmflefac 15994 prmodvdslcmf 16385 prmolelcmf 16386 prmgaplcmlem1 16389 prmgaplcmlem2 16390 prmgaplcm 16398 wilthlem2 25648 wilthlem3 25649 swrdrn2 30630 swrdrn3 30631 swrdf1 30632 cycpmfv2 30758 freshmansdream 30861 fsum2dsub 31880 breprexplema 31903 breprexplemc 31905 breprexpnat 31907 vtsprod 31912 circlemeth 31913 fzisoeu 41574 fzsscn 41585 fzssre 41588 fzct 41655 sumnnodd 41918 dvnprodlem1 42238 dvnprodlem2 42239 fourierdlem20 42419 fourierdlem25 42424 fourierdlem37 42436 fourierdlem52 42450 fourierdlem64 42462 fourierdlem79 42477 etransclem32 42558 |
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