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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 9328 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | uzid 9333 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (ℤ≥‘𝑁)) |
4 | eluzfz 9794 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ (𝑀...𝑁)) | |
5 | 3, 4 | mpdan 417 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ‘cfv 5118 (class class class)co 5767 ℤcz 9047 ℤ≥cuz 9319 ...cfz 9783 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-neg 7929 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: eluzfz2b 9806 elfzubelfz 9809 fzopth 9834 fzsuc 9842 fseq1p1m1 9867 fzm1 9873 fzneuz 9874 fzoend 9992 exfzdc 10010 uzsinds 10208 seq3clss 10233 seq3fveq2 10235 seq3shft2 10239 monoord 10242 monoord2 10243 seq3split 10245 seq3caopr3 10247 seq3f1olemp 10268 seq3id3 10273 seq3id2 10275 ser3ge0 10283 seq3coll 10578 summodclem2a 11143 fsumm1 11178 telfsumo 11228 telfsumo2 11229 fsumparts 11232 prodfap0 11307 prodfrecap 11308 supfz 13226 |
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