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Mathbox for Glauco Siliprandi |
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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 11266 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
5 | xrltnle 11283 | . . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶)) | |
6 | 2, 4, 5 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶)) |
7 | 1, 6 | mpbid 231 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ≤ 𝐶) |
8 | 7 | intnanrd 490 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
9 | ltnelicc.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
10 | elicc4 13393 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
11 | 4, 9, 2, 10 | syl3anc 1371 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (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 ∧ wa 396 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7411 ℝcr 11111 ℝ*cxr 11249 < clt 11250 ≤ cle 11251 [,]cicc 13329 |
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 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-xr 11254 df-le 11256 df-icc 13333 |
This theorem is referenced by: fourierdlem104 45005 |
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