Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrgtnelicc | 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, 3-Jan-2021.) |
Ref | Expression |
---|---|
xrgtnelicc.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrgtnelicc.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrgtnelicc.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
xrgtnelicc.4 | ⊢ (𝜑 → 𝐵 < 𝐶) |
Ref | Expression |
---|---|
xrgtnelicc | ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrgtnelicc.4 | . . . 4 ⊢ (𝜑 → 𝐵 < 𝐶) | |
2 | xrgtnelicc.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | xrgtnelicc.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
4 | xrltnle 10883 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) | |
5 | 2, 3, 4 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) |
6 | 1, 5 | mpbid 235 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵) |
7 | 6 | intnand 492 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
8 | xrgtnelicc.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
9 | elicc4 12985 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
10 | 8, 2, 3, 9 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
11 | 7, 10 | mtbird 328 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 class class class wbr 5043 (class class class)co 7202 ℝ*cxr 10849 < clt 10850 ≤ cle 10851 [,]cicc 12921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-iota 6327 df-fun 6371 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-xr 10854 df-le 10856 df-icc 12925 |
This theorem is referenced by: iccdificc 42704 |
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