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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 9434 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | 1 | simplbi 268 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 ‘cfv 5015 (class class class)co 5652 ℤ≥cuz 9019 ...cfz 9424 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-neg 7656 df-z 8751 df-uz 9020 df-fz 9425 |
This theorem is referenced by: elfzel1 9439 elfzelz 9440 elfzle1 9441 eluzfz2b 9447 fzsplit2 9464 fzsplit 9465 fzopth 9475 fzss1 9477 fzss2 9478 fzssuz 9479 fzp1elp1 9489 uzsplit 9506 elfzmlbm 9542 fzosplit 9588 iseqfeq2 9891 seq3feq2 9893 iseqfeq 9896 isermono 9906 iseqsplit 9908 iseqcaopr3 9910 iseqf1olemkle 9913 iseqf1olemklt 9914 iseqf1olemnab 9917 iseqf1olemqk 9923 iseqf1olemjpcl 9924 iseqf1olemqpcl 9925 iseqf1olemfvp 9926 seq3f1olemqsumkj 9927 seq3f1olemqsumk 9928 seq3f1olemqsum 9929 seq3f1olemstep 9930 seq3f1oleml 9932 seq3f1o 9933 iseqz 9943 iser0 9947 ser0 9949 ser3le 9953 iseqcoll 10247 climub 10733 isumrblem 10765 fisumcvg 10766 fsum3cvg 10767 fisumser 10790 fsump1i 10827 fsum0diaglem 10834 iserabs 10869 isumsplit 10885 isum1p 10886 geosergap 10900 mertenslemi1 10929 infssuzex 11223 prmind2 11380 prmdvdsfz 11398 |
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