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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lbioc | Structured version Visualization version GIF version |
Description: A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
lbioc | ⊢ ¬ 𝐴 ∈ (𝐴(,]𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ioc 13361 | . . . . 5 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
2 | 1 | elixx3g 13369 | . . . 4 ⊢ (𝐴 ∈ (𝐴(,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) ∧ (𝐴 < 𝐴 ∧ 𝐴 ≤ 𝐵))) |
3 | 2 | biimpi 215 | . . 3 ⊢ (𝐴 ∈ (𝐴(,]𝐵) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) ∧ (𝐴 < 𝐴 ∧ 𝐴 ≤ 𝐵))) |
4 | 3 | simprld 771 | . 2 ⊢ (𝐴 ∈ (𝐴(,]𝐵) → 𝐴 < 𝐴) |
5 | 1 | elmpocl1 7663 | . . 3 ⊢ (𝐴 ∈ (𝐴(,]𝐵) → 𝐴 ∈ ℝ*) |
6 | xrltnr 13131 | . . 3 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐴(,]𝐵) → ¬ 𝐴 < 𝐴) |
8 | 4, 7 | pm2.65i 193 | 1 ⊢ ¬ 𝐴 ∈ (𝐴(,]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2099 {crab 3429 class class class wbr 5148 (class class class)co 7420 ℝ*cxr 11277 < clt 11278 ≤ cle 11279 (,]cioc 13357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-ioc 13361 |
This theorem is referenced by: fouriersw 45619 |
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