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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 9568 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | uzid 9573 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (ℤ≥‘𝑁)) |
4 | eluzfz 10052 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ (𝑀...𝑁)) | |
5 | 3, 4 | mpdan 421 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ‘cfv 5235 (class class class)co 5897 ℤcz 9284 ℤ≥cuz 9559 ...cfz 10040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-pre-ltirr 7954 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-neg 8162 df-z 9285 df-uz 9560 df-fz 10041 |
This theorem is referenced by: eluzfz2b 10065 elfzubelfz 10068 fzopth 10093 fzsuc 10101 fseq1p1m1 10126 fzm1 10132 fzneuz 10133 fzoend 10254 exfzdc 10272 uzsinds 10475 seq3clss 10500 seq3fveq2 10502 seq3shft2 10506 monoord 10509 monoord2 10510 seq3split 10512 seq3caopr3 10514 seq3f1olemp 10535 seq3id3 10540 seq3id2 10542 seqfeq4g 10546 ser3ge0 10551 seq3coll 10857 summodclem2a 11424 fsumm1 11459 telfsumo 11509 telfsumo2 11510 fsumparts 11513 prodfap0 11588 prodfrecap 11589 prodmodclem2a 11619 fprodm1 11641 eulerthlemrprm 12264 eulerthlema 12265 nninfdclemlt 12505 gsumval2 12875 supfz 15298 |
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