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Mirrors > Home > MPE Home > Th. List > elfzuz3 | Structured version Visualization version GIF version |
Description: Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzuz3 | ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuzb 13250 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | 1 | simprbi 497 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 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: elfzel2 13254 elfzle2 13260 peano2fzr 13269 fzsplit2 13281 fzsplit 13282 fznn0sub 13288 fzopth 13293 fzss1 13295 fzss2 13296 fzp1elp1 13309 predfz 13381 fzosplit 13420 fzoend 13478 fzofzp1b 13485 uzindi 13702 seqcl2 13741 seqfveq2 13745 monoord 13753 sermono 13755 seqsplit 13756 seqf1olem2 13763 seqid2 13769 seqhomo 13770 seqz 13771 bcval5 14032 seqcoll 14178 seqcoll2 14179 swrdval2 14359 pfxres 14392 pfxf 14393 spllen 14467 splfv2a 14469 repswpfx 14498 fsum0diag2 15495 climcndslem2 15562 prodfn0 15606 lcmflefac 16353 pcbc 16601 vdwlem2 16683 vdwlem5 16686 vdwlem6 16687 vdwlem8 16689 prmgaplem1 16750 psgnunilem5 19102 efgsres 19344 efgredleme 19349 efgcpbllemb 19361 imasdsf1olem 23526 volsup 24720 dvn2bss 25094 dvtaylp 25529 wilth 26220 ftalem1 26222 ppisval2 26254 dvdsppwf1o 26335 logfaclbnd 26370 bposlem6 26437 wlkres 28038 fzsplit3 31115 wrdres 31211 pfxf1 31216 swrdrn2 31226 swrdrn3 31227 swrdf1 31228 swrdrndisj 31229 splfv3 31230 cycpmco2f1 31391 cycpmco2rn 31392 cycpmco2lem7 31399 ballotlemsima 32482 ballotlemfrc 32493 ballotlemfrceq 32495 fzssfzo 32518 signstres 32554 fsum2dsub 32587 revpfxsfxrev 33077 swrdrevpfx 33078 pfxwlk 33085 erdszelem7 33159 erdszelem8 33160 poimirlem1 35778 poimirlem2 35779 poimirlem3 35780 poimirlem4 35781 poimirlem7 35784 poimirlem12 35789 poimirlem15 35792 poimirlem16 35793 poimirlem17 35794 poimirlem19 35796 poimirlem20 35797 poimirlem23 35800 poimirlem24 35801 poimirlem25 35802 poimirlem29 35806 poimirlem31 35808 mettrifi 35915 fzsplitnd 39991 bcc0 41958 iunincfi 42644 monoordxrv 43022 fmulcl 43122 fmul01lt1lem2 43126 dvnprodlem2 43488 stoweidlem11 43552 stoweidlem17 43558 fourierdlem15 43663 ssfz12 44806 smonoord 44823 |
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