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Mirrors > Home > ILE Home > Th. List > fzssp1 | GIF version |
Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fzssp1 | ⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzel2 9772 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
2 | uzid 9308 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
3 | peano2uz 9346 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → (𝑁 + 1) ∈ (ℤ≥‘𝑁)) | |
4 | fzss2 9812 | . . . 4 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑁) → (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) | |
5 | 1, 2, 3, 4 | 4syl 18 | . . 3 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))) |
6 | id 19 | . . 3 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (𝑀...𝑁)) | |
7 | 5, 6 | sseldd 3068 | . 2 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
8 | 7 | ssriv 3071 | 1 ⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 ⊆ wss 3041 ‘cfv 5093 (class class class)co 5742 1c1 7589 + caddc 7591 ℤcz 9022 ℤ≥cuz 9294 ...cfz 9758 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-fz 9759 |
This theorem is referenced by: fzelp1 9822 fseq1p1m1 9842 monoord2 10218 binomlem 11220 binom1dif 11224 |
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