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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 13560 | . . . . 5 ⊢ (𝑘 ∈ (𝐾...𝑁) → 𝑘 ∈ (ℤ≥‘𝐾)) | |
| 2 | id 22 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
| 3 | uztrn 12896 | . . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
| 4 | 1, 2, 3 | syl2anr 597 | . . . 4 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ (𝐾...𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
| 5 | elfzuz3 13561 | . . . . 5 ⊢ (𝑘 ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘𝑘)) | |
| 6 | 5 | adantl 481 | . . . 4 ⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑘 ∈ (𝐾...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑘)) |
| 7 | elfzuzb 13558 | . . . 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 412 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝑘 ∈ (𝐾...𝑁) → 𝑘 ∈ (𝑀...𝑁))) |
| 10 | 9 | ssrdv 3989 | 1 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 ℤ≥cuz 12878 ...cfz 13547 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 |
| This theorem is referenced by: fzssnn 13608 fzp1ss 13615 fzdif1 13645 ige2m1fz 13657 fzoss1 13726 fzossnn0 13730 sermono 14075 seqsplit 14076 seqf1olem2 14083 seqz 14091 seqcoll2 14504 swrdswrd 14743 swrdccatin2 14767 pfxccatin12lem2c 14768 pfxccatpfx2 14775 swrds2m 14980 mertenslem1 15920 reumodprminv 16842 prmgaplcmlem1 17089 structfn 17193 strleun 17194 cpmadugsumlemF 22882 ply1termlem 26242 dvply1 26325 ppisval2 27148 ppiltx 27220 chtlepsi 27250 chtublem 27255 chpub 27264 gausslemma2dlem3 27412 2lgslem1a 27435 chtppilimlem1 27517 pntlemq 27645 pntlemf 27649 axlowdimlem16 28972 axlowdimlem17 28973 axlowdim 28976 cyclnumvtx 29820 crctcshwlkn0lem3 29832 swrdrndisj 32942 esumpmono 34080 ballotlem2 34491 ballotlemfc0 34495 ballotlemfcc 34496 fsum2dsub 34622 chtvalz 34644 poimirlem1 37628 poimirlem2 37629 poimirlem4 37631 poimirlem6 37633 poimirlem7 37634 poimirlem15 37642 poimirlem16 37643 poimirlem19 37646 poimirlem20 37647 poimirlem23 37650 poimirlem27 37654 fdc 37752 jm2.23 43008 stoweidlem11 46026 elaa2lem 46248 iccpartgel 47416 |
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