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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltnelicc | Structured version Visualization version GIF version |
Description: A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ltnelicc.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnelicc.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
ltnelicc.c | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
ltnelicc.clta | ⊢ (𝜑 → 𝐶 < 𝐴) |
Ref | Expression |
---|---|
ltnelicc | ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnelicc.clta | . . . 4 ⊢ (𝜑 → 𝐶 < 𝐴) | |
2 | ltnelicc.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
3 | ltnelicc.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | rexrd 10769 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
5 | xrltnle 10786 | . . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶)) | |
6 | 2, 4, 5 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶)) |
7 | 1, 6 | mpbid 235 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ≤ 𝐶) |
8 | 7 | intnanrd 493 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
9 | ltnelicc.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
10 | elicc4 12888 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
11 | 4, 9, 2, 10 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
12 | 8, 11 | mtbird 328 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 ℝ*cxr 10752 < clt 10753 ≤ cle 10754 [,]cicc 12824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-xr 10757 df-le 10759 df-icc 12828 |
This theorem is referenced by: fourierdlem104 43293 |
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