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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 3967 | . . . 4 ⊢ (𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁)) ↔ (𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁))) | |
| 2 | elfzel1 13563 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
| 3 | 2 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
| 4 | 3 | zred 12722 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ) |
| 5 | elfzel2 13562 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝐾 ∈ ℤ) | |
| 6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) |
| 7 | 6 | zred 12722 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℝ) |
| 8 | elfzelz 13564 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
| 9 | 8 | zred 12722 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
| 10 | 9 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
| 11 | elfzle1 13567 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑥) | |
| 12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑥) |
| 13 | elfzle2 13568 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ≤ 𝐾) | |
| 14 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝐾) |
| 15 | 4, 10, 7, 12, 14 | letrd 11418 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝐾) |
| 16 | 4, 7, 15 | lensymd 11412 | . . . 4 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → ¬ 𝐾 < 𝑀) |
| 17 | 1, 16 | sylbi 217 | . . 3 ⊢ (𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁)) → ¬ 𝐾 < 𝑀) |
| 18 | 17 | con2i 139 | . 2 ⊢ (𝐾 < 𝑀 → ¬ 𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁))) |
| 19 | 18 | eq0rdv 4407 | 1 ⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 < clt 11295 ≤ cle 11296 ℤcz 12613 ...cfz 13547 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 |
| This theorem is referenced by: fsumm1 15787 fsum1p 15789 o1fsum 15849 climcndslem1 15885 climcndslem2 15886 mertenslem1 15920 fprod1p 16004 fprodeq0 16011 fallfacval4 16079 prmreclem5 16958 strleun 17194 uniioombllem3 25620 mtest 26447 birthdaylem2 26995 fsumharmonic 27055 ftalem5 27120 chtdif 27201 ppidif 27206 gausslemma2dlem4 27413 gausslemma2dlem6 27416 lgsquadlem2 27425 dchrisum0lem1b 27559 dchrisum0lem3 27563 pntrsumbnd2 27611 pntrlog2bndlem6 27627 pntpbnd2 27631 pntlemf 27649 axlowdimlem2 28958 axlowdimlem16 28972 esumpmono 34080 ballotlemfrceq 34531 fsum2dsub 34622 poimirlem1 37628 poimirlem2 37629 poimirlem3 37630 poimirlem4 37631 poimirlem6 37633 poimirlem7 37634 poimirlem11 37638 poimirlem12 37639 poimirlem16 37643 poimirlem17 37644 poimirlem19 37646 poimirlem20 37647 poimirlem23 37650 poimirlem24 37651 poimirlem25 37652 poimirlem28 37655 poimirlem29 37656 poimirlem31 37658 sticksstones6 42152 sticksstones7 42153 metakunt18 42223 metakunt20 42225 prodsplit 42241 sumcubes 42347 eldioph2lem1 42771 stoweidlem11 46026 |
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