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Mirrors > Home > ILE Home > Th. List > elfzuz | GIF version |
Description: A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzuz | ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuzb 10005 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | 1 | simplbi 274 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ‘cfv 5212 (class class class)co 5869 ℤ≥cuz 9517 ...cfz 9995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-neg 8121 df-z 9243 df-uz 9518 df-fz 9996 |
This theorem is referenced by: elfzel1 10010 elfzelz 10011 elfzle1 10013 eluzfz2b 10019 fzsplit2 10036 fzsplit 10037 fzopth 10047 fzss1 10049 fzss2 10050 fzssuz 10051 fzp1elp1 10061 uzsplit 10078 elfzmlbm 10117 fzosplit 10163 seq3feq2 10456 seq3feq 10458 ser3mono 10464 seq3caopr3 10467 iseqf1olemkle 10470 iseqf1olemklt 10471 iseqf1olemnab 10474 iseqf1olemqk 10480 iseqf1olemjpcl 10481 iseqf1olemqpcl 10482 iseqf1olemfvp 10483 seq3f1olemqsumkj 10484 seq3f1olemqsumk 10485 seq3f1olemqsum 10486 seq3f1olemstep 10487 seq3f1oleml 10489 seq3f1o 10490 seq3z 10497 ser0 10500 ser3le 10504 seq3coll 10806 climub 11336 sumrbdclem 11369 fsum3cvg 11370 fsum3ser 11389 fsump1i 11425 fsum0diaglem 11432 iserabs 11467 isumsplit 11483 isum1p 11484 geosergap 11498 mertenslemi1 11527 prodf1 11534 prodfap0 11537 prodfrecap 11538 prodfdivap 11539 prodrbdclem 11563 fproddccvg 11564 fprodntrivap 11576 fprodabs 11608 fprodeq0 11609 infssuzex 11933 prmind2 12103 prmdvdsfz 12122 isprm5lem 12124 eulerthlemrprm 12212 eulerthlema 12213 pcfac 12331 lgsdilem2 14104 cvgcmp2nlemabs 14436 |
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