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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 10701 | . . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶)) |
5 | 1, 4 | mpbid 234 | . . 3 ⊢ (𝜑 → ¬ 𝐴 < 𝐶) |
6 | 5 | intn3an2d 1476 | . 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
7 | lenelioc.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
8 | elioc1 12774 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
9 | 3, 7, 8 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
10 | 6, 9 | mtbird 327 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 (,]cioc 12733 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
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 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-xr 10673 df-le 10675 df-ioc 12737 |
This theorem is referenced by: (None) |
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