Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13253 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
2 | eluzelz 12592 | . 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 2106 ‘cfv 6433 (class class class)co 7275 ℤcz 12319 ℤ≥cuz 12582 ...cfz 13239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-neg 11208 df-z 12320 df-uz 12583 df-fz 13240 |
This theorem is referenced by: elfz1eq 13267 fzdisj 13283 fzssp1 13299 fzp1disj 13315 fzrev2i 13321 fzrev3 13322 elfz1b 13325 fznuz 13338 fznn0sub2 13363 elfzmlbm 13366 difelfznle 13370 nn0disj 13372 fz1fzo0m1 13435 fzofzp1b 13485 bcm1k 14029 bcp1nk 14031 pfxccatin12lem2 14444 spllen 14467 fsum0diag2 15495 fallfacval3 15722 fallfacval4 15753 psgnunilem2 19103 pntpbnd1 26734 crctcshwlkn0 28186 fzm1ne1 31110 swrdrevpfx 33078 swrdwlk 33088 elfzfzo 42815 sumnnodd 43171 dvnmul 43484 dvnprodlem1 43487 dvnprodlem2 43488 stoweidlem34 43575 fourierdlem11 43659 fourierdlem12 43660 fourierdlem15 43663 fourierdlem41 43689 fourierdlem48 43695 fourierdlem49 43696 fourierdlem54 43701 fourierdlem79 43726 fourierdlem102 43749 fourierdlem114 43761 etransclem23 43798 etransclem35 43810 iundjiun 43998 2elfz2melfz 44810 elfzelfzlble 44813 iccpartiltu 44874 iccpartgt 44879 |
Copyright terms: Public domain | W3C validator |