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Mirrors > Home > MPE Home > Th. List > fzss1 | Structured version Visualization version GIF version |
Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fzss1 | ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz 12892 | . . . . 5 ⊢ (𝑘 ∈ (𝐾...𝑁) → 𝑘 ∈ (ℤ≥‘𝐾)) | |
2 | id 22 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
3 | uztrn 12249 | . . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
4 | 1, 2, 3 | syl2anr 596 | . . . 4 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ (𝐾...𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
5 | elfzuz3 12893 | . . . . 5 ⊢ (𝑘 ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘𝑘)) | |
6 | 5 | adantl 482 | . . . 4 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ (𝐾...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑘)) |
7 | elfzuzb 12890 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) ↔ (𝑘 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑘))) | |
8 | 4, 6, 7 | sylanbrc 583 | . . 3 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ (𝐾...𝑁)) → 𝑘 ∈ (𝑀...𝑁)) |
9 | 8 | ex 413 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝑘 ∈ (𝐾...𝑁) → 𝑘 ∈ (𝑀...𝑁))) |
10 | 9 | ssrdv 3970 | 1 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 ℤ≥cuz 12231 ...cfz 12880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-neg 10861 df-z 11970 df-uz 12232 df-fz 12881 |
This theorem is referenced by: fzssnn 12939 fzp1ss 12946 ige2m1fz 12985 fzoss1 13052 fzossnn0 13056 sermono 13390 seqsplit 13391 seqf1olem2 13398 seqz 13406 seqcoll2 13811 swrdswrd 14055 swrdccatin2 14079 pfxccatin12lem2c 14080 pfxccatpfx2 14087 swrds2m 14291 mertenslem1 15228 reumodprminv 16129 prmgaplcmlem1 16375 structfn 16488 strleun 16579 cpmadugsumlemF 21412 ply1termlem 24720 dvply1 24800 ppisval2 25609 ppiltx 25681 chtlepsi 25709 chtublem 25714 chpub 25723 gausslemma2dlem3 25871 2lgslem1a 25894 chtppilimlem1 25976 pntlemq 26104 pntlemf 26108 axlowdimlem16 26670 axlowdimlem17 26671 axlowdim 26674 crctcshwlkn0lem3 27517 swrdrndisj 30558 esumpmono 31237 ballotlem2 31645 ballotlemfc0 31649 ballotlemfcc 31650 fsum2dsub 31777 chtvalz 31799 poimirlem1 34774 poimirlem2 34775 poimirlem4 34777 poimirlem6 34779 poimirlem7 34780 poimirlem15 34788 poimirlem16 34789 poimirlem19 34792 poimirlem20 34793 poimirlem23 34796 poimirlem27 34800 fdc 34901 jm2.23 39471 stoweidlem11 42173 elaa2lem 42395 iccpartgel 43466 |
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