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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 3927 | . . . 4 ⊢ (𝑥 ∈ ((𝐽...𝐾) ∩ (𝑀...𝑁)) ↔ (𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁))) | |
2 | elfzel1 13446 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
3 | 2 | adantl 483 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
4 | 3 | zred 12612 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ) |
5 | elfzel2 13445 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝐾 ∈ ℤ) | |
6 | 5 | adantr 482 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) |
7 | 6 | zred 12612 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℝ) |
8 | elfzelz 13447 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
9 | 8 | zred 12612 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
10 | 9 | adantl 483 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
11 | elfzle1 13450 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑥) | |
12 | 11 | adantl 483 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑥) |
13 | elfzle2 13451 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽...𝐾) → 𝑥 ≤ 𝐾) | |
14 | 13 | adantr 482 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝐾) |
15 | 4, 10, 7, 12, 14 | letrd 11317 | . . . . 5 ⊢ ((𝑥 ∈ (𝐽...𝐾) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝐾) |
16 | 4, 7, 15 | lensymd 11311 | . . . 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 4365 | 1 ⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ∅c0 4283 class class class wbr 5106 (class class class)co 7358 ℝcr 11055 < clt 11194 ≤ cle 11195 ℤcz 12504 ...cfz 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-neg 11393 df-z 12505 df-uz 12769 df-fz 13431 |
This theorem is referenced by: fsumm1 15641 fsum1p 15643 o1fsum 15703 climcndslem1 15739 climcndslem2 15740 mertenslem1 15774 fprod1p 15856 fprodeq0 15863 fallfacval4 15931 prmreclem5 16797 strleun 17034 uniioombllem3 24965 mtest 25779 birthdaylem2 26318 fsumharmonic 26377 ftalem5 26442 chtdif 26523 ppidif 26528 gausslemma2dlem4 26733 gausslemma2dlem6 26736 lgsquadlem2 26745 dchrisum0lem1b 26879 dchrisum0lem3 26883 pntrsumbnd2 26931 pntrlog2bndlem6 26947 pntpbnd2 26951 pntlemf 26969 axlowdimlem2 27934 axlowdimlem16 27948 esumpmono 32735 ballotlemfrceq 33185 fsum2dsub 33277 poimirlem1 36125 poimirlem2 36126 poimirlem3 36127 poimirlem4 36128 poimirlem6 36130 poimirlem7 36131 poimirlem11 36135 poimirlem12 36136 poimirlem16 36140 poimirlem17 36141 poimirlem19 36143 poimirlem20 36144 poimirlem23 36147 poimirlem24 36148 poimirlem25 36149 poimirlem28 36152 poimirlem29 36153 poimirlem31 36155 sticksstones6 40605 sticksstones7 40606 metakunt18 40640 metakunt20 40642 prodsplit 40659 eldioph2lem1 41126 stoweidlem11 44338 |
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