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Mirrors > Home > ILE Home > Th. List > fzdisj | Unicode version |
Description: Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Ref | Expression |
---|---|
fzdisj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3120 |
. . . 4
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2 | elfzel1 8659 |
. . . . . . . 8
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3 | 2 | adantl 262 |
. . . . . . 7
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4 | 3 | zred 8136 |
. . . . . 6
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5 | elfzelz 8660 |
. . . . . . . 8
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6 | 5 | zred 8136 |
. . . . . . 7
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7 | 6 | adantl 262 |
. . . . . 6
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8 | elfzel2 8658 |
. . . . . . . 8
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9 | 8 | adantr 261 |
. . . . . . 7
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10 | 9 | zred 8136 |
. . . . . 6
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11 | elfzle1 8661 |
. . . . . . 7
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12 | 11 | adantl 262 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | elfzle2 8662 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | adantr 261 |
. . . . . 6
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15 | 4, 7, 10, 12, 14 | letrd 6935 |
. . . . 5
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16 | 4, 10 | lenltd 6931 |
. . . . 5
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17 | 15, 16 | mpbid 135 |
. . . 4
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18 | 1, 17 | sylbi 114 |
. . 3
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19 | 18 | con2i 557 |
. 2
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20 | 19 | eq0rdv 3255 |
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 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-pre-ltwlin 6796 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-neg 6982 df-z 8022 df-uz 8250 df-fz 8645 |
This theorem is referenced by: (None) |
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