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Mirrors > Home > MPE Home > Th. List > elfzel2 | Structured version Visualization version GIF version |
Description: Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzel2 | ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz3 12893 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
2 | eluzelz 12241 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ ℤ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-neg 10861 df-z 11970 df-uz 12232 df-fz 12881 |
This theorem is referenced by: elfz1eq 12906 fzdisj 12922 fzssp1 12938 fzp1disj 12954 fzrev2i 12960 fzrev3 12961 elfz1b 12964 fznuz 12977 fznn0sub2 13002 elfzmlbm 13005 difelfznle 13009 nn0disj 13011 fz1fzo0m1 13073 fzofzp1b 13123 bcm1k 13663 bcp1nk 13665 pfxccatin12lem2 14081 spllen 14104 fsum0diag2 15126 fallfacval3 15354 fallfacval4 15385 psgnunilem2 18552 pntpbnd1 26089 crctcshwlkn0 27526 fzm1ne1 30438 swrdrevpfx 32260 swrdwlk 32270 elfzfzo 41418 sumnnodd 41787 dvnmul 42104 dvnprodlem1 42107 dvnprodlem2 42108 stoweidlem34 42196 fourierdlem11 42280 fourierdlem12 42281 fourierdlem15 42284 fourierdlem41 42310 fourierdlem48 42316 fourierdlem49 42317 fourierdlem54 42322 fourierdlem79 42347 fourierdlem102 42370 fourierdlem114 42382 etransclem23 42419 etransclem35 42431 iundjiun 42619 2elfz2melfz 43395 elfzelfzlble 43398 iccpartiltu 43459 iccpartgt 43464 |
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