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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 10428 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
4 | gtnelioc.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
5 | xrltnle 10446 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) | |
6 | 3, 4, 5 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) |
7 | 1, 6 | mpbid 224 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵) |
8 | 7 | intn3an3d 1554 | . 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
9 | gtnelioc.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
10 | elioc2 12552 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
11 | 9, 2, 10 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
12 | 8, 11 | mtbird 317 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ w3a 1071 ∈ wcel 2107 class class class wbr 4888 (class class class)co 6924 ℝcr 10273 ℝ*cxr 10412 < clt 10413 ≤ cle 10414 (,]cioc 12492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-pre-lttri 10348 ax-pre-lttrn 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-ioc 12496 |
This theorem is referenced by: fourierswlem 41384 fouriersw 41385 etransclem18 41406 etransclem46 41434 |
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