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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 13112 | . 2 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
2 | 1 | ssriv 3905 | 1 ⊢ (𝑀...𝑁) ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3866 (class class class)co 7213 ℤcz 12176 ...cfz 13095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-neg 11065 df-z 12177 df-uz 12439 df-fz 13096 |
This theorem is referenced by: fzof 13240 fzossz 13262 seqcoll 14030 lcmflefac 16205 prmodvdslcmf 16600 prmolelcmf 16601 prmgaplcmlem1 16604 prmgaplcmlem2 16605 prmgaplcm 16613 wilthlem2 25951 wilthlem3 25952 cycpmfv2 31100 freshmansdream 31203 breprexplema 32322 breprexplemc 32324 breprexpnat 32326 vtsprod 32331 lcmfunnnd 39754 lcmineqlem4 39774 fzisoeu 42512 fzsscn 42523 fzssre 42526 fzct 42591 dvnprodlem1 43162 dvnprodlem2 43163 fourierdlem20 43343 fourierdlem25 43348 fourierdlem37 43360 fourierdlem52 43374 fourierdlem64 43386 fourierdlem79 43401 |
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