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Mirrors > Home > MPE Home > Th. List > fzdisj | Structured version Visualization version GIF version |
Description: Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Ref | Expression |
---|---|
fzdisj | ⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3963 | . . . 4 ⊢ (𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁)) ↔ (𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁))) | |
2 | elfzel1 13504 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
3 | 2 | adantl 480 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
4 | 3 | zred 12670 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ) |
5 | elfzel2 13503 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝐾 ∈ ℤ) | |
6 | 5 | adantr 479 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) |
7 | 6 | zred 12670 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℝ) |
8 | elfzelz 13505 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
9 | 8 | zred 12670 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
10 | 9 | adantl 480 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
11 | elfzle1 13508 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑥) | |
12 | 11 | adantl 480 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑥) |
13 | elfzle2 13509 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ≤ 𝐾) | |
14 | 13 | adantr 479 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝐾) |
15 | 4, 10, 7, 12, 14 | letrd 11375 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝐾) |
16 | 4, 7, 15 | lensymd 11369 | . . . 4 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → ¬ 𝐾 < 𝑀) |
17 | 1, 16 | sylbi 216 | . . 3 ⊢ (𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁)) → ¬ 𝐾 < 𝑀) |
18 | 17 | con2i 139 | . 2 ⊢ (𝐾 < 𝑀 → ¬ 𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁))) |
19 | 18 | eq0rdv 4403 | 1 ⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∩ cin 3946 ∅c0 4321 class class class wbr 5147 (class class class)co 7411 ℝcr 11111 < clt 11252 ≤ cle 11253 ℤcz 12562 ...cfz 13488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 df-fz 13489 |
This theorem is referenced by: fsumm1 15701 fsum1p 15703 o1fsum 15763 climcndslem1 15799 climcndslem2 15800 mertenslem1 15834 fprod1p 15916 fprodeq0 15923 fallfacval4 15991 prmreclem5 16857 strleun 17094 uniioombllem3 25334 mtest 26152 birthdaylem2 26693 fsumharmonic 26752 ftalem5 26817 chtdif 26898 ppidif 26903 gausslemma2dlem4 27108 gausslemma2dlem6 27111 lgsquadlem2 27120 dchrisum0lem1b 27254 dchrisum0lem3 27258 pntrsumbnd2 27306 pntrlog2bndlem6 27322 pntpbnd2 27326 pntlemf 27344 axlowdimlem2 28468 axlowdimlem16 28482 esumpmono 33375 ballotlemfrceq 33825 fsum2dsub 33917 poimirlem1 36792 poimirlem2 36793 poimirlem3 36794 poimirlem4 36795 poimirlem6 36797 poimirlem7 36798 poimirlem11 36802 poimirlem12 36803 poimirlem16 36807 poimirlem17 36808 poimirlem19 36810 poimirlem20 36811 poimirlem23 36814 poimirlem24 36815 poimirlem25 36816 poimirlem28 36819 poimirlem29 36820 poimirlem31 36822 sticksstones6 41273 sticksstones7 41274 metakunt18 41308 metakunt20 41310 prodsplit 41327 sumcubes 41513 eldioph2lem1 41800 stoweidlem11 45025 |
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