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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 13561 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
| 2 | eluzelz 12888 | . 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 2108 ‘cfv 6561 (class class class)co 7431 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 | 
| This theorem is referenced by: elfz1eq 13575 fzdisj 13591 fzssp1 13607 fzp1disj 13623 fzrev2i 13629 fzrev3 13630 elfz1b 13633 fznuz 13649 fznn0sub2 13675 elfzmlbm 13678 difelfznle 13682 nn0disj 13684 fz1fzo0m1 13750 fzofzp1b 13804 bcm1k 14354 bcp1nk 14356 pfxccatin12lem2 14769 spllen 14792 fsum0diag2 15819 fallfacval3 16048 fallfacval4 16079 psgnunilem2 19513 pntpbnd1 27630 crctcshwlkn0 29841 fzm1ne1 32790 swrdrevpfx 35122 swrdwlk 35132 elfzfzo 45288 sumnnodd 45645 dvnmul 45958 dvnprodlem1 45961 dvnprodlem2 45962 stoweidlem34 46049 fourierdlem11 46133 fourierdlem12 46134 fourierdlem15 46137 fourierdlem41 46163 fourierdlem48 46169 fourierdlem49 46170 fourierdlem54 46175 fourierdlem79 46200 fourierdlem102 46223 fourierdlem114 46235 etransclem23 46272 etransclem35 46284 iundjiun 46475 2elfz2melfz 47330 elfzelfzlble 47333 iccpartiltu 47409 iccpartgt 47414 | 
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