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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 12336 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | 1 | simprbi 480 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℤ≥cuz 11687 ...cfz 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-neg 10269 df-z 11378 df-uz 11688 df-fz 12327 |
This theorem is referenced by: elfzel2 12340 elfzle2 12345 peano2fzr 12354 fzsplit2 12366 fzsplit 12367 fznn0sub 12373 fzopth 12378 fzss1 12380 fzss2 12381 fzp1elp1 12394 predfz 12464 fzosplit 12501 fzoend 12559 fzofzp1b 12566 uzindi 12781 seqcl2 12819 seqfveq2 12823 monoord 12831 sermono 12833 seqsplit 12834 seqf1olem2 12841 seqid2 12847 seqhomo 12848 seqz 12849 bcval5 13105 seqcoll 13248 seqcoll2 13249 swrdval2 13420 swrd0val 13421 swrd0len 13422 spllen 13505 splfv2a 13507 fsum0diag2 14515 climcndslem2 14582 prodfn0 14626 lcmflefac 15361 pcbc 15604 vdwlem2 15686 vdwlem5 15689 vdwlem6 15690 vdwlem8 15692 prmgaplem1 15753 psgnunilem5 17914 efgsres 18151 efgredleme 18156 efgcpbllemb 18168 imasdsf1olem 22178 volsup 23324 dvn2bss 23693 dvtaylp 24124 wilth 24797 ftalem1 24799 ppisval2 24831 dvdsppwf1o 24912 logfaclbnd 24947 bposlem6 25014 wlkres 26567 fzsplit3 29553 ballotlemsima 30577 ballotlemfrc 30588 ballotlemfrceq 30590 fzssfzo 30613 wrdres 30617 signstres 30652 fsum2dsub 30685 erdszelem7 31179 erdszelem8 31180 poimirlem1 33410 poimirlem2 33411 poimirlem3 33412 poimirlem4 33413 poimirlem7 33416 poimirlem12 33421 poimirlem15 33424 poimirlem16 33425 poimirlem17 33426 poimirlem19 33428 poimirlem20 33429 poimirlem23 33432 poimirlem24 33433 poimirlem25 33434 poimirlem29 33438 poimirlem31 33440 mettrifi 33553 bcc0 38539 iunincfi 39272 fmulcl 39813 fmul01lt1lem2 39817 dvnprodlem2 40162 stoweidlem11 40228 stoweidlem17 40234 fourierdlem15 40339 ssfz12 41324 smonoord 41341 pfxres 41388 pfxf 41389 repswpfx 41436 |
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