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Mirrors > Home > ILE Home > Th. List > fznlem | Unicode version |
Description: A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.) |
Ref | Expression |
---|---|
fznlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 8249 | . . . . . . . . . . 11 | |
2 | zre 8249 | . . . . . . . . . . 11 | |
3 | lenlt 7094 | . . . . . . . . . . 11 | |
4 | 1, 2, 3 | syl2an 273 | . . . . . . . . . 10 |
5 | 4 | biimpd 132 | . . . . . . . . 9 |
6 | 5 | con2d 554 | . . . . . . . 8 |
7 | 6 | imp 115 | . . . . . . 7 |
8 | 7 | adantr 261 | . . . . . 6 |
9 | simplll 485 | . . . . . . . 8 | |
10 | 9 | zred 8360 | . . . . . . 7 |
11 | simpr 103 | . . . . . . . 8 | |
12 | 11 | zred 8360 | . . . . . . 7 |
13 | simpllr 486 | . . . . . . . 8 | |
14 | 13 | zred 8360 | . . . . . . 7 |
15 | letr 7101 | . . . . . . 7 | |
16 | 10, 12, 14, 15 | syl3anc 1135 | . . . . . 6 |
17 | 8, 16 | mtod 589 | . . . . 5 |
18 | 17 | ralrimiva 2392 | . . . 4 |
19 | rabeq0 3247 | . . . 4 | |
20 | 18, 19 | sylibr 137 | . . 3 |
21 | fzval 8876 | . . . . 5 | |
22 | 21 | eqeq1d 2048 | . . . 4 |
23 | 22 | adantr 261 | . . 3 |
24 | 20, 23 | mpbird 156 | . 2 |
25 | 24 | ex 108 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 crab 2310 c0 3224 class class class wbr 3764 (class class class)co 5512 cr 6888 clt 7060 cle 7061 cz 8245 cfz 8874 |
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-13 1404 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-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltwlin 6997 |
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-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 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-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-neg 7185 df-z 8246 df-fz 8875 |
This theorem is referenced by: fzn 8906 |
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