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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 9483 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | uzid 9488 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (ℤ≥‘𝑁)) |
4 | eluzfz 9963 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ (𝑀...𝑁)) | |
5 | 3, 4 | mpdan 419 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 ‘cfv 5196 (class class class)co 5850 ℤcz 9199 ℤ≥cuz 9474 ...cfz 9952 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-pre-ltirr 7873 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-neg 8080 df-z 9200 df-uz 9475 df-fz 9953 |
This theorem is referenced by: eluzfz2b 9976 elfzubelfz 9979 fzopth 10004 fzsuc 10012 fseq1p1m1 10037 fzm1 10043 fzneuz 10044 fzoend 10165 exfzdc 10183 uzsinds 10385 seq3clss 10410 seq3fveq2 10412 seq3shft2 10416 monoord 10419 monoord2 10420 seq3split 10422 seq3caopr3 10424 seq3f1olemp 10445 seq3id3 10450 seq3id2 10452 ser3ge0 10460 seq3coll 10764 summodclem2a 11331 fsumm1 11366 telfsumo 11416 telfsumo2 11417 fsumparts 11420 prodfap0 11495 prodfrecap 11496 prodmodclem2a 11526 fprodm1 11548 eulerthlemrprm 12170 eulerthlema 12171 nninfdclemlt 12393 supfz 14022 |
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