![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9831 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | 1 | simplbi 272 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ‘cfv 5131 (class class class)co 5782 ℤ≥cuz 9350 ...cfz 9821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-neg 7960 df-z 9079 df-uz 9351 df-fz 9822 |
This theorem is referenced by: elfzel1 9836 elfzelz 9837 elfzle1 9838 eluzfz2b 9844 fzsplit2 9861 fzsplit 9862 fzopth 9872 fzss1 9874 fzss2 9875 fzssuz 9876 fzp1elp1 9886 uzsplit 9903 elfzmlbm 9939 fzosplit 9985 seq3feq2 10274 seq3feq 10276 ser3mono 10282 seq3caopr3 10285 iseqf1olemkle 10288 iseqf1olemklt 10289 iseqf1olemnab 10292 iseqf1olemqk 10298 iseqf1olemjpcl 10299 iseqf1olemqpcl 10300 iseqf1olemfvp 10301 seq3f1olemqsumkj 10302 seq3f1olemqsumk 10303 seq3f1olemqsum 10304 seq3f1olemstep 10305 seq3f1oleml 10307 seq3f1o 10308 seq3z 10315 ser0 10318 ser3le 10322 seq3coll 10617 climub 11145 sumrbdclem 11178 fsum3cvg 11179 fsum3ser 11198 fsump1i 11234 fsum0diaglem 11241 iserabs 11276 isumsplit 11292 isum1p 11293 geosergap 11307 mertenslemi1 11336 prodf1 11343 prodfap0 11346 prodfrecap 11347 prodfdivap 11348 prodrbdclem 11372 fproddccvg 11373 infssuzex 11678 prmind2 11837 prmdvdsfz 11855 cvgcmp2nlemabs 13402 |
Copyright terms: Public domain | W3C validator |