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Mirrors > Home > ILE Home > Th. List > eluzgtdifelfzo | Unicode version |
Description: Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
Ref | Expression |
---|---|
eluzgtdifelfzo |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 |
. . . . 5
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2 | 1 | adantl 271 |
. . . 4
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3 | simpl 107 |
. . . . . 6
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4 | 3 | adantr 270 |
. . . . 5
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5 | eluzelz 9028 |
. . . . . . . 8
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6 | 5 | ad2antrr 472 |
. . . . . . 7
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7 | simprr 499 |
. . . . . . 7
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8 | 6, 7 | zsubcld 8873 |
. . . . . 6
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9 | 8 | ancoms 264 |
. . . . 5
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10 | 4, 9 | zaddcld 8872 |
. . . 4
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11 | zre 8754 |
. . . . . . . . 9
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12 | zre 8754 |
. . . . . . . . 9
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13 | posdif 7933 |
. . . . . . . . . 10
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14 | 13 | biimpd 142 |
. . . . . . . . 9
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15 | 11, 12, 14 | syl2anr 284 |
. . . . . . . 8
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16 | 15 | adantld 272 |
. . . . . . 7
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17 | 16 | imp 122 |
. . . . . 6
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18 | resubcl 7746 |
. . . . . . . . 9
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19 | 12, 11, 18 | syl2an 283 |
. . . . . . . 8
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20 | 19 | adantr 270 |
. . . . . . 7
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21 | eluzelre 9029 |
. . . . . . . 8
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22 | 21 | ad2antrl 474 |
. . . . . . 7
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23 | 20, 22 | ltaddposd 8006 |
. . . . . 6
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24 | 17, 23 | mpbid 145 |
. . . . 5
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25 | zcn 8755 |
. . . . . . 7
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26 | 25 | ad2antrr 472 |
. . . . . 6
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27 | eluzelcn 9030 |
. . . . . . 7
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28 | 27 | ad2antrl 474 |
. . . . . 6
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29 | zcn 8755 |
. . . . . . . 8
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30 | 29 | adantl 271 |
. . . . . . 7
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31 | 30 | adantr 270 |
. . . . . 6
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32 | addsub12 7695 |
. . . . . . 7
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33 | 32 | breq2d 3857 |
. . . . . 6
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34 | 26, 28, 31, 33 | syl3anc 1174 |
. . . . 5
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35 | 24, 34 | mpbird 165 |
. . . 4
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36 | elfzo2 9561 |
. . . 4
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37 | 2, 10, 35, 36 | syl3anbrc 1127 |
. . 3
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38 | fzosubel3 9607 |
. . 3
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39 | 37, 9, 38 | syl2anc 403 |
. 2
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40 | 39 | ex 113 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-addcom 7445 ax-addass 7447 ax-distr 7449 ax-i2m1 7450 ax-0lt1 7451 ax-0id 7453 ax-rnegex 7454 ax-cnre 7456 ax-pre-ltirr 7457 ax-pre-ltwlin 7458 ax-pre-lttrn 7459 ax-pre-ltadd 7461 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-pnf 7524 df-mnf 7525 df-xr 7526 df-ltxr 7527 df-le 7528 df-sub 7655 df-neg 7656 df-inn 8423 df-n0 8674 df-z 8751 df-uz 9020 df-fz 9425 df-fzo 9554 |
This theorem is referenced by: ige2m2fzo 9609 |
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