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Mirrors > Home > MPE Home > Th. List > fzdisj | Structured version Visualization version 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 3616 |
. . . 4
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2 | elfzel1 11796 |
. . . . . . . 8
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3 | 2 | adantl 468 |
. . . . . . 7
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4 | 3 | zred 11037 |
. . . . . 6
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5 | elfzelz 11797 |
. . . . . . . 8
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6 | 5 | zred 11037 |
. . . . . . 7
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7 | 6 | adantl 468 |
. . . . . 6
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8 | elfzel2 11795 |
. . . . . . . 8
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9 | 8 | adantr 467 |
. . . . . . 7
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10 | 9 | zred 11037 |
. . . . . 6
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11 | elfzle1 11799 |
. . . . . . 7
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12 | 11 | adantl 468 |
. . . . . 6
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13 | elfzle2 11800 |
. . . . . . 7
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14 | 13 | adantr 467 |
. . . . . 6
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15 | 4, 7, 10, 12, 14 | letrd 9789 |
. . . . 5
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16 | 4, 10 | lenltd 9778 |
. . . . 5
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17 | 15, 16 | mpbid 214 |
. . . 4
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18 | 1, 17 | sylbi 199 |
. . 3
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19 | 18 | con2i 124 |
. 2
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20 | 19 | eq0rdv 3768 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-cnex 9592 ax-resscn 9593 ax-pre-lttri 9610 ax-pre-lttrn 9611 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-1st 6790 df-2nd 6791 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-neg 9860 df-z 10935 df-uz 11157 df-fz 11782 |
This theorem is referenced by: fsumm1 13805 fsum1p 13807 o1fsum 13866 climcndslem1 13900 climcndslem2 13901 mertenslem1 13933 fprod1p 14015 fprodeq0 14022 fallfacval4 14089 prmreclem5 14857 strleun 15213 uniioombllem3 22536 mtest 23352 birthdaylem2 23871 fsumharmonic 23930 ftalem5 23994 ftalem5OLD 23996 chtdif 24078 ppidif 24083 lgsquadlem2 24276 dchrisum0lem1b 24346 dchrisum0lem3 24350 pntrsumbnd2 24398 pntrlog2bndlem6 24414 pntpbnd2 24418 pntlemf 24436 axlowdimlem2 24966 axlowdimlem16 24980 constr3trllem3 25373 esumpmono 28893 ballotlemfrceq 29354 ballotlemfrceqOLD 29392 poimirlem1 31934 poimirlem2 31935 poimirlem3 31936 poimirlem4 31937 poimirlem6 31939 poimirlem7 31940 poimirlem11 31944 poimirlem12 31945 poimirlem16 31949 poimirlem17 31950 poimirlem19 31952 poimirlem20 31953 poimirlem23 31956 poimirlem24 31957 poimirlem25 31958 poimirlem28 31961 poimirlem29 31962 poimirlem31 31964 eldioph2lem1 35596 stoweidlem11 37865 |
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