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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 10679 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
4 | gtnelicc.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
5 | xrltnle 10696 | . . . . 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 233 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵) |
8 | 7 | intnand 489 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
9 | gtnelicc.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
10 | elicc4 12791 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
11 | 9, 3, 4, 10 | syl3anc 1363 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
12 | 8, 11 | mtbird 326 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-xr 10667 df-le 10669 df-icc 12733 |
This theorem is referenced by: fourierdlem103 42371 |
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