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Mirrors > Home > ILE Home > Th. List > elznn0 | Unicode version |
Description: Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
elznn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9014 | . 2 | |
2 | elnn0 8937 | . . . . . 6 | |
3 | 2 | a1i 9 | . . . . 5 |
4 | elnn0 8937 | . . . . . 6 | |
5 | recn 7721 | . . . . . . . . 9 | |
6 | 0cn 7726 | . . . . . . . . 9 | |
7 | negcon1 7982 | . . . . . . . . 9 | |
8 | 5, 6, 7 | sylancl 409 | . . . . . . . 8 |
9 | neg0 7976 | . . . . . . . . . 10 | |
10 | 9 | eqeq1i 2125 | . . . . . . . . 9 |
11 | eqcom 2119 | . . . . . . . . 9 | |
12 | 10, 11 | bitri 183 | . . . . . . . 8 |
13 | 8, 12 | syl6bb 195 | . . . . . . 7 |
14 | 13 | orbi2d 764 | . . . . . 6 |
15 | 4, 14 | syl5bb 191 | . . . . 5 |
16 | 3, 15 | orbi12d 767 | . . . 4 |
17 | 3orass 950 | . . . . 5 | |
18 | orcom 702 | . . . . 5 | |
19 | orordir 748 | . . . . 5 | |
20 | 17, 18, 19 | 3bitrri 206 | . . . 4 |
21 | 16, 20 | syl6rbb 196 | . . 3 |
22 | 21 | pm5.32i 449 | . 2 |
23 | 1, 22 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wo 682 w3o 946 wceq 1316 wcel 1465 cc 7586 cr 7587 cc0 7588 cneg 7902 cn 8684 cn0 8935 cz 9012 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-neg 7904 df-n0 8936 df-z 9013 |
This theorem is referenced by: peano2z 9048 zmulcl 9065 elz2 9080 expnegzap 10282 expaddzaplem 10291 odd2np1 11482 bezoutlemzz 11602 bezoutlemaz 11603 bezoutlembz 11604 |
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