Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 9968 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
5 | xrltnle 9984 | . . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶)) | |
6 | 2, 4, 5 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶)) |
7 | 1, 6 | mpbid 221 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ≤ 𝐶) |
8 | 7 | intnanrd 954 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
9 | ltnelicc.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
10 | elicc4 12111 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
11 | 4, 9, 2, 10 | syl3anc 1318 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
12 | 8, 11 | mtbird 314 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-xr 9957 df-le 9959 df-icc 12053 |
This theorem is referenced by: fourierdlem104 39103 |
Copyright terms: Public domain | W3C validator |