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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 13555 | . 2 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
2 | 1 | ssriv 3983 | 1 ⊢ (𝑀...𝑁) ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3947 (class class class)co 7424 ℤcz 12610 ...cfz 13538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-neg 11497 df-z 12611 df-uz 12875 df-fz 13539 |
This theorem is referenced by: fzof 13683 fzossz 13706 seqcoll 14483 lcmflefac 16649 prmodvdslcmf 17049 prmolelcmf 17050 prmgaplcmlem1 17053 prmgaplcmlem2 17054 prmgaplcm 17062 freshmansdream 21572 wilthlem2 27097 wilthlem3 27098 cycpmfv2 32992 breprexplema 34476 breprexplemc 34478 breprexpnat 34480 vtsprod 34485 lcmfunnnd 41711 lcmineqlem4 41731 aks6d1c6lem5 41875 fzisoeu 44915 fzsscn 44926 fzssre 44929 fzct 44994 dvnprodlem1 45567 dvnprodlem2 45568 fourierdlem20 45748 fourierdlem25 45753 fourierdlem37 45765 fourierdlem52 45779 fourierdlem64 45791 fourierdlem79 45806 |
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