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Mirrors > Home > ILE Home > Th. List > fzssuz | 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 9833 | . 2 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
2 | 1 | ssriv 3106 | 1 ⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀) |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3076 ‘cfv 5131 (class class class)co 5782 ℤ≥cuz 9350 ...cfz 9821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-neg 7960 df-z 9079 df-uz 9351 df-fz 9822 |
This theorem is referenced by: fzssnn 9879 fzossnn0 9983 seq3split 10283 seq3caopr2 10286 summodclem2a 11182 fisumss 11193 fsumsersdc 11196 isumclim3 11224 binomlem 11284 prodmodclem2a 11377 isprm3 11835 2prm 11844 |
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