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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 8754 |
. . . . . . . . . . 11
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2 | zre 8754 |
. . . . . . . . . . 11
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3 | lenlt 7561 |
. . . . . . . . . . 11
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4 | 1, 2, 3 | syl2an 283 |
. . . . . . . . . 10
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5 | 4 | biimpd 142 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 5 | con2d 589 |
. . . . . . . 8
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7 | 6 | imp 122 |
. . . . . . 7
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8 | 7 | adantr 270 |
. . . . . 6
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9 | simplll 500 |
. . . . . . . 8
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10 | 9 | zred 8868 |
. . . . . . 7
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11 | simpr 108 |
. . . . . . . 8
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12 | 11 | zred 8868 |
. . . . . . 7
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13 | simpllr 501 |
. . . . . . . 8
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14 | 13 | zred 8868 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | letr 7568 |
. . . . . . 7
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16 | 10, 12, 14, 15 | syl3anc 1174 |
. . . . . 6
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17 | 8, 16 | mtod 624 |
. . . . 5
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18 | 17 | ralrimiva 2446 |
. . . 4
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19 | rabeq0 3312 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 18, 19 | sylibr 132 |
. . 3
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21 | fzval 9426 |
. . . . 5
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22 | 21 | eqeq1d 2096 |
. . . 4
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23 | 22 | adantr 270 |
. . 3
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24 | 20, 23 | mpbird 165 |
. 2
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25 | 24 | 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-pre-ltwlin 7458 |
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-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-pnf 7524 df-mnf 7525 df-xr 7526 df-ltxr 7527 df-le 7528 df-neg 7656 df-z 8751 df-fz 9425 |
This theorem is referenced by: fzn 9456 |
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