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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 4019 | . . . 4 ⊢ (𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁)) ↔ (𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁))) | |
2 | elfzel1 12658 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
3 | 2 | adantl 475 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
4 | 3 | zred 11834 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ) |
5 | elfzel2 12657 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝐾 ∈ ℤ) | |
6 | 5 | adantr 474 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) |
7 | 6 | zred 11834 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℝ) |
8 | elfzelz 12659 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
9 | 8 | zred 11834 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
10 | 9 | adantl 475 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
11 | elfzle1 12661 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑥) | |
12 | 11 | adantl 475 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑥) |
13 | elfzle2 12662 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ≤ 𝐾) | |
14 | 13 | adantr 474 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝐾) |
15 | 4, 10, 7, 12, 14 | letrd 10533 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝐾) |
16 | 4, 7, 15 | lensymd 10527 | . . . 4 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → ¬ 𝐾 < 𝑀) |
17 | 1, 16 | sylbi 209 | . . 3 ⊢ (𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁)) → ¬ 𝐾 < 𝑀) |
18 | 17 | con2i 137 | . 2 ⊢ (𝐾 < 𝑀 → ¬ 𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁))) |
19 | 18 | eq0rdv 4205 | 1 ⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ∅c0 4141 class class class wbr 4886 (class class class)co 6922 ℝcr 10271 < clt 10411 ≤ cle 10412 ℤcz 11728 ...cfz 12643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-neg 10609 df-z 11729 df-uz 11993 df-fz 12644 |
This theorem is referenced by: fsumm1 14887 fsum1p 14889 o1fsum 14949 climcndslem1 14985 climcndslem2 14986 mertenslem1 15019 fprod1p 15101 fprodeq0 15108 fallfacval4 15176 prmreclem5 16028 strleun 16364 uniioombllem3 23789 mtest 24595 birthdaylem2 25131 fsumharmonic 25190 ftalem5 25255 chtdif 25336 ppidif 25341 gausslemma2dlem4 25546 gausslemma2dlem6 25549 lgsquadlem2 25558 dchrisum0lem1b 25656 dchrisum0lem3 25660 pntrsumbnd2 25708 pntrlog2bndlem6 25724 pntpbnd2 25728 pntlemf 25746 axlowdimlem2 26292 axlowdimlem16 26306 esumpmono 30739 ballotlemfrceq 31189 fsum2dsub 31287 poimirlem1 34036 poimirlem2 34037 poimirlem3 34038 poimirlem4 34039 poimirlem6 34041 poimirlem7 34042 poimirlem11 34046 poimirlem12 34047 poimirlem16 34051 poimirlem17 34052 poimirlem19 34054 poimirlem20 34055 poimirlem23 34058 poimirlem24 34059 poimirlem25 34060 poimirlem28 34063 poimirlem29 34064 poimirlem31 34066 eldioph2lem1 38283 stoweidlem11 41155 |
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