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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lenelioc | Structured version Visualization version GIF version |
Description: A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
lenelioc.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
lenelioc.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
lenelioc.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
lenelioc.4 | ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
Ref | Expression |
---|---|
lenelioc | ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lenelioc.4 | . . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴) | |
2 | lenelioc.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
3 | lenelioc.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | 2, 3 | xrlenltd 10445 | . . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶)) |
5 | 1, 4 | mpbid 224 | . . 3 ⊢ (𝜑 → ¬ 𝐴 < 𝐶) |
6 | 5 | intn3an2d 1553 | . 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
7 | lenelioc.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
8 | elioc1 12534 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
9 | 3, 7, 8 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
10 | 6, 9 | mtbird 317 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ w3a 1071 ∈ wcel 2107 class class class wbr 4888 (class class class)co 6924 ℝ*cxr 10412 < clt 10413 ≤ cle 10414 (,]cioc 12493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-iota 6101 df-fun 6139 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-xr 10417 df-le 10419 df-ioc 12497 |
This theorem is referenced by: (None) |
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