![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3964 | . . . 4 ⊢ (𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁)) ↔ (𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁))) | |
2 | elfzel1 13507 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
3 | 2 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
4 | 3 | zred 12673 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ) |
5 | elfzel2 13506 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝐾 ∈ ℤ) | |
6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) |
7 | 6 | zred 12673 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℝ) |
8 | elfzelz 13508 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
9 | 8 | zred 12673 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
10 | 9 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
11 | elfzle1 13511 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑥) | |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑥) |
13 | elfzle2 13512 | . . . . . . 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 11378 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝐾) |
16 | 4, 7, 15 | lensymd 11372 | . . . 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 4404 | 1 ⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∩ cin 3947 ∅c0 4322 class class class wbr 5148 (class class class)co 7412 ℝcr 11115 < clt 11255 ≤ cle 11256 ℤcz 12565 ...cfz 13491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-pre-lttri 11190 ax-pre-lttrn 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-neg 11454 df-z 12566 df-uz 12830 df-fz 13492 |
This theorem is referenced by: fsumm1 15704 fsum1p 15706 o1fsum 15766 climcndslem1 15802 climcndslem2 15803 mertenslem1 15837 fprod1p 15919 fprodeq0 15926 fallfacval4 15994 prmreclem5 16860 strleun 17097 uniioombllem3 25347 mtest 26166 birthdaylem2 26708 fsumharmonic 26767 ftalem5 26832 chtdif 26913 ppidif 26918 gausslemma2dlem4 27123 gausslemma2dlem6 27126 lgsquadlem2 27135 dchrisum0lem1b 27269 dchrisum0lem3 27273 pntrsumbnd2 27321 pntrlog2bndlem6 27337 pntpbnd2 27341 pntlemf 27359 axlowdimlem2 28483 axlowdimlem16 28497 esumpmono 33390 ballotlemfrceq 33840 fsum2dsub 33932 poimirlem1 36805 poimirlem2 36806 poimirlem3 36807 poimirlem4 36808 poimirlem6 36810 poimirlem7 36811 poimirlem11 36815 poimirlem12 36816 poimirlem16 36820 poimirlem17 36821 poimirlem19 36823 poimirlem20 36824 poimirlem23 36827 poimirlem24 36828 poimirlem25 36829 poimirlem28 36832 poimirlem29 36833 poimirlem31 36835 sticksstones6 41286 sticksstones7 41287 metakunt18 41321 metakunt20 41323 prodsplit 41340 sumcubes 41526 eldioph2lem1 41813 stoweidlem11 45038 |
Copyright terms: Public domain | W3C validator |