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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gtnelioc | Structured version Visualization version GIF version |
Description: A real number larger than the upper bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
gtnelioc.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
gtnelioc.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
gtnelioc.c | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
gtnelioc.bltc | ⊢ (𝜑 → 𝐵 < 𝐶) |
Ref | Expression |
---|---|
gtnelioc | ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gtnelioc.bltc | . . . 4 ⊢ (𝜑 → 𝐵 < 𝐶) | |
2 | gtnelioc.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 2 | rexrd 11260 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
4 | gtnelioc.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
5 | xrltnle 11277 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) | |
6 | 3, 4, 5 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) |
7 | 1, 6 | mpbid 231 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵) |
8 | 7 | intn3an3d 1481 | . 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
9 | gtnelioc.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
10 | elioc2 13383 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
11 | 9, 2, 10 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
12 | 8, 11 | mtbird 324 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5147 (class class class)co 7405 ℝcr 11105 ℝ*cxr 11243 < clt 11244 ≤ cle 11245 (,]cioc 13321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ioc 13325 |
This theorem is referenced by: fourierswlem 44932 fouriersw 44933 etransclem18 44954 etransclem46 44982 |
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