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Mirrors > Home > ILE Home > Th. List > eluzfz2 | Unicode 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 9566 |
. . 3
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2 | uzid 9571 |
. . 3
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3 | 1, 2 | syl 14 |
. 2
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4 | eluzfz 10049 |
. 2
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5 | 3, 4 | mpdan 421 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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 7931 ax-resscn 7932 ax-pre-ltirr 7952 |
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 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-neg 8160 df-z 9283 df-uz 9558 df-fz 10038 |
This theorem is referenced by: eluzfz2b 10062 elfzubelfz 10065 fzopth 10090 fzsuc 10098 fseq1p1m1 10123 fzm1 10129 fzneuz 10130 fzoend 10251 exfzdc 10269 uzsinds 10472 seq3clss 10497 seq3fveq2 10499 seq3shft2 10503 monoord 10506 monoord2 10507 seq3split 10509 seq3caopr3 10511 seq3f1olemp 10532 seq3id3 10537 seq3id2 10539 ser3ge0 10547 seq3coll 10853 summodclem2a 11420 fsumm1 11455 telfsumo 11505 telfsumo2 11506 fsumparts 11509 prodfap0 11584 prodfrecap 11585 prodmodclem2a 11615 fprodm1 11637 eulerthlemrprm 12260 eulerthlema 12261 nninfdclemlt 12501 supfz 15273 |
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