| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fzssuz | Structured version Visualization version GIF version | ||
| Description: A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.) |
| Ref | Expression |
|---|---|
| fzssuz | ⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzuz 13560 | . 2 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
| 2 | 1 | ssriv 3987 | 1 ⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀) |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ 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-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| 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-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-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 |
| This theorem is referenced by: ltwefz 14004 seqcoll2 14504 caubnd 15397 climsup 15706 summolem2a 15751 fsumss 15761 fsumsers 15764 isumclim3 15795 binomlem 15865 prodmolem2a 15970 fprodntriv 15978 fprodss 15984 iprodclim3 16036 fprodefsum 16131 isprm3 16720 2prm 16729 prmreclem5 16958 4sqlem11 16993 gsumval3 19925 telgsums 20011 fz2ssnn0 32787 elrgspnlem2 33247 esumpcvgval 34079 esumcvg 34087 eulerpartlemsv3 34363 ballotlemfc0 34495 ballotlemfcc 34496 ballotlemiex 34504 ballotlemsima 34518 ballotlemrv2 34524 fsum2dsub 34622 erdszelem4 35199 erdszelem8 35203 volsupnfl 37672 sdclem2 37749 geomcau 37766 diophin 42783 irrapxlem1 42833 fzssnn0 45329 iuneqfzuzlem 45345 fzossuz 45392 uzublem 45441 climinf 45621 sge0uzfsumgt 46459 iundjiun 46475 caratheodorylem1 46541 |
| Copyright terms: Public domain | W3C validator |