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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 9427 |
. . 3
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2 | uzid 9432 |
. . 3
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3 | 1, 2 | syl 14 |
. 2
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4 | eluzfz 9901 |
. 2
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5 | 3, 4 | mpdan 418 |
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 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 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-cnex 7802 ax-resscn 7803 ax-pre-ltirr 7823 |
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 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-rab 2441 df-v 2711 df-sbc 2934 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-br 3962 df-opab 4022 df-mpt 4023 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-fv 5171 df-ov 5817 df-oprab 5818 df-mpo 5819 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-neg 8028 df-z 9147 df-uz 9419 df-fz 9891 |
This theorem is referenced by: eluzfz2b 9913 elfzubelfz 9916 fzopth 9941 fzsuc 9949 fseq1p1m1 9974 fzm1 9980 fzneuz 9981 fzoend 10099 exfzdc 10117 uzsinds 10319 seq3clss 10344 seq3fveq2 10346 seq3shft2 10350 monoord 10353 monoord2 10354 seq3split 10356 seq3caopr3 10358 seq3f1olemp 10379 seq3id3 10384 seq3id2 10386 ser3ge0 10394 seq3coll 10690 summodclem2a 11255 fsumm1 11290 telfsumo 11340 telfsumo2 11341 fsumparts 11344 prodfap0 11419 prodfrecap 11420 prodmodclem2a 11450 fprodm1 11472 supfz 13580 |
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