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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 10693 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
5 | xrltnle 10710 | . . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶)) | |
6 | 2, 4, 5 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶)) |
7 | 1, 6 | mpbid 234 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ≤ 𝐶) |
8 | 7 | intnanrd 492 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
9 | ltnelicc.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
10 | elicc4 12806 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
11 | 4, 9, 2, 10 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
12 | 8, 11 | mtbird 327 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 [,]cicc 12744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-xr 10681 df-le 10683 df-icc 12748 |
This theorem is referenced by: fourierdlem104 42502 |
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