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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 10707 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) | |
5 | 2, 3, 4 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) |
6 | 1, 5 | mpbid 234 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵) |
7 | 6 | intnand 491 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
8 | xrgtnelicc.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
9 | elicc4 12802 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
10 | 8, 2, 3, 9 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
11 | 7, 10 | mtbird 327 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 class class class wbr 5065 (class class class)co 7155 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 [,]cicc 12740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-xr 10678 df-le 10680 df-icc 12744 |
This theorem is referenced by: iccdificc 41813 |
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