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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 10105 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) | |
5 | 2, 3, 4 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵)) |
6 | 1, 5 | mpbid 222 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵) |
7 | 6 | intnand 962 | . 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
8 | xrgtnelicc.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
9 | elicc4 12240 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
10 | 8, 2, 3, 9 | syl3anc 1326 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
11 | 7, 10 | mtbird 315 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 [,]cicc 12178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-xr 10078 df-le 10080 df-icc 12182 |
This theorem is referenced by: iccdificc 39766 |
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