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Mirrors > Home > ILE Home > Th. List > eluzfz2 | GIF version |
Description: Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
eluzfz2 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 8482 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | uzid 8487 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (ℤ≥‘𝑁)) |
4 | eluzfz 8885 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ (𝑀...𝑁)) | |
5 | 3, 4 | mpdan 398 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ‘cfv 4902 (class class class)co 5512 ℤcz 8245 ℤ≥cuz 8473 ...cfz 8874 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-neg 7185 df-z 8246 df-uz 8474 df-fz 8875 |
This theorem is referenced by: eluzfz2b 8897 elfzubelfz 8900 fzopth 8924 fzsuc 8931 fseq1p1m1 8956 fzm1 8962 fzneuz 8963 fzoend 9078 iseqfveq2 9228 iseqshft2 9232 monoord 9235 monoord2 9236 iseqsplit 9238 iseqcaopr3 9240 iseqid3s 9246 |
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