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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 9359 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | uzid 9364 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (ℤ≥‘𝑁)) |
4 | eluzfz 9832 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ (𝑀...𝑁)) | |
5 | 3, 4 | mpdan 418 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ‘cfv 5131 (class class class)co 5782 ℤcz 9078 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-13 1492 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-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 |
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-nel 2405 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-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-neg 7960 df-z 9079 df-uz 9351 df-fz 9822 |
This theorem is referenced by: eluzfz2b 9844 elfzubelfz 9847 fzopth 9872 fzsuc 9880 fseq1p1m1 9905 fzm1 9911 fzneuz 9912 fzoend 10030 exfzdc 10048 uzsinds 10246 seq3clss 10271 seq3fveq2 10273 seq3shft2 10277 monoord 10280 monoord2 10281 seq3split 10283 seq3caopr3 10285 seq3f1olemp 10306 seq3id3 10311 seq3id2 10313 ser3ge0 10321 seq3coll 10617 summodclem2a 11182 fsumm1 11217 telfsumo 11267 telfsumo2 11268 fsumparts 11271 prodfap0 11346 prodfrecap 11347 prodmodclem2a 11377 supfz 13428 |
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