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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 12592 | . . . . 5 ⊢ (𝑘 ∈ (𝐾...𝑁) → 𝑘 ∈ (ℤ≥‘𝐾)) | |
2 | id 22 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
3 | uztrn 11947 | . . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
4 | 1, 2, 3 | syl2anr 591 | . . . 4 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ (𝐾...𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
5 | elfzuz3 12593 | . . . . 5 ⊢ (𝑘 ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘𝑘)) | |
6 | 5 | adantl 474 | . . . 4 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ (𝐾...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑘)) |
7 | elfzuzb 12590 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) ↔ (𝑘 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑘))) | |
8 | 4, 6, 7 | sylanbrc 579 | . . 3 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ (𝐾...𝑁)) → 𝑘 ∈ (𝑀...𝑁)) |
9 | 8 | ex 402 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝑘 ∈ (𝐾...𝑁) → 𝑘 ∈ (𝑀...𝑁))) |
10 | 9 | ssrdv 3804 | 1 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ⊆ wss 3769 ‘cfv 6101 (class class class)co 6878 ℤ≥cuz 11930 ...cfz 12580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-pre-lttri 10298 ax-pre-lttrn 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-neg 10559 df-z 11667 df-uz 11931 df-fz 12581 |
This theorem is referenced by: fzssnn 12639 fzp1ss 12646 ige2m1fz 12684 fzoss1 12750 fzossnn0 12754 sermono 13087 seqsplit 13088 seqf1olem2 13095 seqz 13103 seqcoll2 13498 swrdswrd 13748 swrdccatin2 13790 swrdccatin12lem2c 13791 pfxccatin12 13795 swrdccatin12OLD 13796 pfxccatpfx2 13802 swrds2m 14026 mertenslem1 14953 reumodprminv 15842 prmgaplcmlem1 16088 structfn 16201 strleun 16293 cpmadugsumlemF 21009 ply1termlem 24300 dvply1 24380 ppisval2 25183 ppiltx 25255 chtlepsi 25283 chtublem 25288 chpub 25297 gausslemma2dlem3 25445 2lgslem1a 25468 chtppilimlem1 25514 pntlemq 25642 pntlemf 25646 axlowdimlem16 26194 axlowdimlem17 26195 axlowdim 26198 crctcshwlkn0lem3 27063 esumpmono 30657 ballotlem2 31067 ballotlemfc0 31071 ballotlemfcc 31072 fsum2dsub 31205 chtvalz 31227 poimirlem1 33899 poimirlem2 33900 poimirlem4 33902 poimirlem6 33904 poimirlem7 33905 poimirlem15 33913 poimirlem16 33914 poimirlem19 33917 poimirlem20 33918 poimirlem23 33921 poimirlem27 33925 fdc 34028 jm2.23 38344 stoweidlem11 40967 elaa2lem 41189 iccpartgel 42201 |
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