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Mirrors > Home > MPE Home > Th. List > fzss1 | Structured version Visualization version Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz 12338 | . . . . 5 | |
2 | id 22 | . . . . 5 | |
3 | uztrn 11704 | . . . . 5 | |
4 | 1, 2, 3 | syl2anr 495 | . . . 4 |
5 | elfzuz3 12339 | . . . . 5 | |
6 | 5 | adantl 482 | . . . 4 |
7 | elfzuzb 12336 | . . . 4 | |
8 | 4, 6, 7 | sylanbrc 698 | . . 3 |
9 | 8 | ex 450 | . 2 |
10 | 9 | ssrdv 3609 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 wss 3574 cfv 5888 (class class class)co 6650 cuz 11687 cfz 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-neg 10269 df-z 11378 df-uz 11688 df-fz 12327 |
This theorem is referenced by: fzssnn 12385 fzp1ss 12392 ige2m1fz 12430 fzoss1 12495 fzossnn0 12499 sermono 12833 seqsplit 12834 seqf1olem2 12841 seqz 12849 bcpasc 13108 seqcoll2 13249 swrd0fv0 13440 swrd0fvlsw 13443 swrdswrd 13460 swrdccatin2 13487 swrdccatin12lem2c 13488 swrdccatin12 13491 mertenslem1 14616 reumodprminv 15509 prmgaplcmlem1 15755 structfn 15874 strleun 15972 efgsres 18151 efgredlemd 18157 efgredlem 18160 chfacfpmmulgsum2 20670 cpmadugsumlemF 20681 ply1termlem 23959 dvply1 24039 dvtaylp 24124 taylthlem2 24128 basellem5 24811 ppisval2 24831 ppiltx 24903 chtlepsi 24931 chtublem 24936 chpub 24945 gausslemma2dlem3 25093 2lgslem1a 25116 chtppilimlem1 25162 pntlemq 25290 pntlemf 25294 axlowdimlem16 25837 axlowdimlem17 25838 axlowdim 25841 crctcshwlkn0lem3 26704 esumpmono 30141 ballotlem2 30550 ballotlemfc0 30554 ballotlemfcc 30555 ballotlemfrci 30589 ballotlemfrceq 30590 fsum2dsub 30685 chtvalz 30707 bcprod 31624 poimirlem1 33410 poimirlem2 33411 poimirlem4 33413 poimirlem6 33415 poimirlem7 33416 poimirlem14 33423 poimirlem15 33424 poimirlem16 33425 poimirlem19 33428 poimirlem20 33429 poimirlem23 33432 poimirlem27 33436 poimirlem31 33440 poimirlem32 33441 fdc 33541 jm2.23 37563 stoweidlem11 40228 elaa2lem 40450 iccpartgel 41365 pfxfv0 41400 pfxfvlsw 41403 pfxccatin12 41425 pfxccatpfx2 41428 |
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