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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 11180 | . . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶)) |
5 | 1, 4 | mpbid 231 | . . 3 ⊢ (𝜑 → ¬ 𝐴 < 𝐶) |
6 | 5 | intn3an2d 1481 | . 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
7 | lenelioc.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
8 | elioc1 13261 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
9 | 3, 7, 8 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
10 | 6, 9 | mtbird 325 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5104 (class class class)co 7352 ℝ*cxr 11147 < clt 11148 ≤ cle 11149 (,]cioc 13220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6446 df-fun 6496 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-xr 11152 df-le 11154 df-ioc 13224 |
This theorem is referenced by: (None) |
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