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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 3198 |
. . . 4
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2 | elfzel1 9588 |
. . . . . . . 8
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3 | 2 | adantl 272 |
. . . . . . 7
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4 | 3 | zred 8967 |
. . . . . 6
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5 | elfzelz 9589 |
. . . . . . . 8
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6 | 5 | zred 8967 |
. . . . . . 7
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7 | 6 | adantl 272 |
. . . . . 6
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8 | elfzel2 9587 |
. . . . . . . 8
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9 | 8 | adantr 271 |
. . . . . . 7
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10 | 9 | zred 8967 |
. . . . . 6
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11 | elfzle1 9590 |
. . . . . . 7
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12 | 11 | adantl 272 |
. . . . . 6
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13 | elfzle2 9591 |
. . . . . . 7
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14 | 13 | adantr 271 |
. . . . . 6
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15 | 4, 7, 10, 12, 14 | letrd 7704 |
. . . . 5
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16 | 4, 10 | lenltd 7698 |
. . . . 5
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17 | 15, 16 | mpbid 146 |
. . . 4
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18 | 1, 17 | sylbi 120 |
. . 3
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19 | 18 | con2i 595 |
. 2
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20 | 19 | eq0rdv 3346 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-pre-ltwlin 7555 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-neg 7753 df-z 8849 df-uz 9119 df-fz 9574 |
This theorem is referenced by: fsumm1 10959 fsum1p 10961 mertenslemi1 11078 strleund 11731 strleun 11732 |
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