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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 8247 | . 2 | |
2 | elnn0 8183 | . . . . . 6 | |
3 | 2 | a1i 9 | . . . . 5 |
4 | elnn0 8183 | . . . . . 6 | |
5 | recn 7014 | . . . . . . . . 9 | |
6 | 0cn 7019 | . . . . . . . . 9 | |
7 | negcon1 7263 | . . . . . . . . 9 | |
8 | 5, 6, 7 | sylancl 392 | . . . . . . . 8 |
9 | neg0 7257 | . . . . . . . . . 10 | |
10 | 9 | eqeq1i 2047 | . . . . . . . . 9 |
11 | eqcom 2042 | . . . . . . . . 9 | |
12 | 10, 11 | bitri 173 | . . . . . . . 8 |
13 | 8, 12 | syl6bb 185 | . . . . . . 7 |
14 | 13 | orbi2d 704 | . . . . . 6 |
15 | 4, 14 | syl5bb 181 | . . . . 5 |
16 | 3, 15 | orbi12d 707 | . . . 4 |
17 | 3orass 888 | . . . . 5 | |
18 | orcom 647 | . . . . 5 | |
19 | orordir 691 | . . . . 5 | |
20 | 17, 18, 19 | 3bitrri 196 | . . . 4 |
21 | 16, 20 | syl6rbb 186 | . . 3 |
22 | 21 | pm5.32i 427 | . 2 |
23 | 1, 22 | bitri 173 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wo 629 w3o 884 wceq 1243 wcel 1393 cc 6887 cr 6888 cc0 6889 cneg 7183 cn 7914 cn0 8181 cz 8245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-neg 7185 df-n0 8182 df-z 8246 |
This theorem is referenced by: peano2z 8281 zmulcl 8297 elz2 8312 expnegzap 9289 expaddzaplem 9298 |
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