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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 9216 | . . . . . . . . . . 11 | |
2 | zre 9216 | . . . . . . . . . . 11 | |
3 | lenlt 7995 | . . . . . . . . . . 11 | |
4 | 1, 2, 3 | syl2an 287 | . . . . . . . . . 10 |
5 | 4 | biimpd 143 | . . . . . . . . 9 |
6 | 5 | con2d 619 | . . . . . . . 8 |
7 | 6 | imp 123 | . . . . . . 7 |
8 | 7 | adantr 274 | . . . . . 6 |
9 | simplll 528 | . . . . . . . 8 | |
10 | 9 | zred 9334 | . . . . . . 7 |
11 | simpr 109 | . . . . . . . 8 | |
12 | 11 | zred 9334 | . . . . . . 7 |
13 | simpllr 529 | . . . . . . . 8 | |
14 | 13 | zred 9334 | . . . . . . 7 |
15 | letr 8002 | . . . . . . 7 | |
16 | 10, 12, 14, 15 | syl3anc 1233 | . . . . . 6 |
17 | 8, 16 | mtod 658 | . . . . 5 |
18 | 17 | ralrimiva 2543 | . . . 4 |
19 | rabeq0 3444 | . . . 4 | |
20 | 18, 19 | sylibr 133 | . . 3 |
21 | fzval 9967 | . . . . 5 | |
22 | 21 | eqeq1d 2179 | . . . 4 |
23 | 22 | adantr 274 | . . 3 |
24 | 20, 23 | mpbird 166 | . 2 |
25 | 24 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 crab 2452 c0 3414 class class class wbr 3989 (class class class)co 5853 cr 7773 clt 7954 cle 7955 cz 9212 cfz 9965 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltwlin 7887 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-neg 8093 df-z 9213 df-fz 9966 |
This theorem is referenced by: fzn 9998 |
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