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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 13538 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
2 | eluzelz 12870 | . 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 2098 ‘cfv 6549 (class class class)co 7419 ℤcz 12596 ℤ≥cuz 12860 ...cfz 13524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-neg 11484 df-z 12597 df-uz 12861 df-fz 13525 |
This theorem is referenced by: elfz1eq 13552 fzdisj 13568 fzssp1 13584 fzp1disj 13600 fzrev2i 13606 fzrev3 13607 elfz1b 13610 fznuz 13623 fznn0sub2 13648 elfzmlbm 13651 difelfznle 13655 nn0disj 13657 fz1fzo0m1 13720 fzofzp1b 13771 bcm1k 14315 bcp1nk 14317 pfxccatin12lem2 14722 spllen 14745 fsum0diag2 15770 fallfacval3 15997 fallfacval4 16028 psgnunilem2 19467 pntpbnd1 27569 crctcshwlkn0 29709 fzm1ne1 32644 swrdrevpfx 34859 swrdwlk 34869 elfzfzo 44798 sumnnodd 45158 dvnmul 45471 dvnprodlem1 45474 dvnprodlem2 45475 stoweidlem34 45562 fourierdlem11 45646 fourierdlem12 45647 fourierdlem15 45650 fourierdlem41 45676 fourierdlem48 45682 fourierdlem49 45683 fourierdlem54 45688 fourierdlem79 45713 fourierdlem102 45736 fourierdlem114 45748 etransclem23 45785 etransclem35 45797 iundjiun 45988 2elfz2melfz 46838 elfzelfzlble 46841 iccpartiltu 46901 iccpartgt 46906 |
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