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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gtnelicc | Structured version Visualization version GIF version |
Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
gtnelicc.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
gtnelicc.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
gtnelicc.c | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
gtnelicc.bltc | ⊢ (𝜑 → 𝐵 < 𝐶) |
Ref | Expression |
---|---|
gtnelicc | ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gtnelicc.bltc | . . . 4 ⊢ (𝜑 → 𝐵 < 𝐶) | |
2 | gtnelicc.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 2 | rexrd 10406 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
4 | gtnelicc.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
5 | xrltnle 10424 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) | |
6 | 3, 4, 5 | syl2anc 581 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) |
7 | 1, 6 | mpbid 224 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵) |
8 | 7 | intnand 484 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
9 | gtnelicc.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
10 | elicc4 12528 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
11 | 9, 3, 4, 10 | syl3anc 1496 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (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 ∧ wa 386 ∈ wcel 2166 class class class wbr 4873 (class class class)co 6905 ℝcr 10251 ℝ*cxr 10390 < clt 10391 ≤ cle 10392 [,]cicc 12466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-xr 10395 df-le 10397 df-icc 12470 |
This theorem is referenced by: fourierdlem103 41220 |
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