P. 104 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xsubge0 10301	Extended real version of subge0 8811. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +e -e𝐵) ↔ 𝐵 ≤ 𝐴))
Theorem	xposdif 10302	Extended real version of posdif 8791. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +e -e𝐴)))
Theorem	xlesubadd 10303	Under certain conditions, the conclusion of lesubadd 8770 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (0 ≤ 𝐴 ∧ 𝐵 ≠ -∞ ∧ 0 ≤ 𝐶)) → ((𝐴 +e -e𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 +e 𝐵)))
Theorem	xaddcld 10304	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐴 +e 𝐵) ∈ ℝ*)
Theorem	xadd4d 10305	Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8503. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞))    &   ⊢ (𝜑 → (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞))    &   ⊢ (𝜑 → (𝐶 ∈ ℝ* ∧ 𝐶 ≠ -∞))    &   ⊢ (𝜑 → (𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞))    ⇒   ⊢ (𝜑 → ((𝐴 +e 𝐵) +e (𝐶 +e 𝐷)) = ((𝐴 +e 𝐶) +e (𝐵 +e 𝐷)))
Theorem	xnn0add4d 10306	Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 10305. (Contributed by AV, 12-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℕ0*)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0*)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0*)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0*)    ⇒   ⊢ (𝜑 → ((𝐴 +e 𝐵) +e (𝐶 +e 𝐷)) = ((𝐴 +e 𝐶) +e (𝐵 +e 𝐷)))
Theorem	xleaddadd 10307	Cancelling a factor of two in ≤ (expressed as addition rather than as a factor to avoid extended real multiplication). (Contributed by Jim Kingdon, 18-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 +e 𝐴) ≤ (𝐵 +e 𝐵)))
4.5.3  Real number intervals
Syntax	cioo 10308	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 10309	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 10310	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 10311	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 10312*	Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
Definition	df-ioc 10313*	Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
Definition	df-ico 10314*	Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
Definition	df-icc 10315*	Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
Theorem	ixxval 10316*	Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
Theorem	elixx1 10317*	Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
Theorem	ixxf 10318*	The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
Theorem	ixxex 10319*	The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ 𝑂 ∈ V
Theorem	ixxssxr 10320*	The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ (𝐴𝑂𝐵) ⊆ ℝ*
Theorem	elixx3g 10321*	Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ* and 𝐵 ∈ ℝ*. (Contributed by Mario Carneiro, 3-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
Theorem	ixxssixx 10322*	An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})    &   ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤))    &   ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))    ⇒   ⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)
Theorem	ixxdisj 10323*	Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})    &   ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) = ∅)
Theorem	ixxss1 10324*	Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤))    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))
Theorem	ixxss2 10325*	Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑇𝑦)})    &   ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))    ⇒   ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))
Theorem	ixxss12 10326*	Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})    &   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐶 ∧ 𝐶𝑇𝑤) → 𝐴𝑅𝑤))    &   ⊢ ((𝑤 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤𝑈𝐷 ∧ 𝐷𝑋𝐵) → 𝑤𝑆𝐵))    ⇒   ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴𝑊𝐶 ∧ 𝐷𝑋𝐵)) → (𝐶𝑃𝐷) ⊆ (𝐴𝑂𝐵))
Theorem	iooex 10327	The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (,) ∈ V
Theorem	iooval 10328*	Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
Theorem	iooidg 10329	An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
⊢ (𝐴 ∈ ℝ* → (𝐴(,)𝐴) = ∅)
Theorem	elioo3g 10330	Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ* and 𝐵 ∈ ℝ*. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	elioo1 10331	Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	elioore 10332	A member of an open interval of reals is a real. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (𝐴 ∈ (𝐵(,)𝐶) → 𝐴 ∈ ℝ)
Theorem	lbioog 10333	An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ 𝐴 ∈ (𝐴(,)𝐵))
Theorem	ubioog 10334	An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴(,)𝐵))
Theorem	iooval2 10335*	Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
Theorem	iooss1 10336	Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
Theorem	iooss2 10337	Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
Theorem	iocval 10338*	Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)})
Theorem	icoval 10339*	Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)})
Theorem	iccval 10340*	Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)})
Theorem	elioo2 10341	Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	elioc1 10342	Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
Theorem	elico1 10343	Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	elicc1 10344	Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
Theorem	iccid 10345	A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
⊢ (𝐴 ∈ ℝ* → (𝐴[,]𝐴) = {𝐴})
Theorem	icc0r 10346	An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → (𝐴[,]𝐵) = ∅))
Theorem	eliooxr 10347	An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*))
Theorem	eliooord 10348	Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.)
⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐴 ∧ 𝐴 < 𝐶))
Theorem	ubioc1 10349	The upper bound belongs to an open-below, closed-above interval. See ubicc2 10405. (Contributed by FL, 29-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵))
Theorem	lbico1 10350	The lower bound belongs to a closed-below, open-above interval. See lbicc2 10404. (Contributed by FL, 29-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
Theorem	iccleub 10351	An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
Theorem	iccgelb 10352	An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
Theorem	elioo5 10353	Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	elioo4g 10354	Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	ioossre 10355	An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
⊢ (𝐴(,)𝐵) ⊆ ℝ
Theorem	elioc2 10356	Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
Theorem	elico2 10357	Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	elicc2 10358	Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
Theorem	elicc2i 10359	Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
Theorem	elicc4 10360	Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
Theorem	iccss 10361	Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
Theorem	iccssioo 10362	Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
Theorem	icossico 10363	Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴[,)𝐵))
Theorem	iccss2 10364	Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
Theorem	iccssico 10365	Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))
Theorem	iccssioo2 10366	Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
Theorem	iccssico2 10367	Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))
Theorem	ioomax 10368	The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
⊢ (-∞(,)+∞) = ℝ
Theorem	iccmax 10369	The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
⊢ (-∞[,]+∞) = ℝ*
Theorem	ioopos 10370	The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
Theorem	ioorp 10371	The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ (0(,)+∞) = ℝ+
Theorem	iooshf 10372	Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 − 𝐵) ∈ (𝐶(,)𝐷) ↔ 𝐴 ∈ ((𝐶 + 𝐵)(,)(𝐷 + 𝐵))))
Theorem	iocssre 10373	A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
Theorem	icossre 10374	A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ⊆ ℝ)
Theorem	iccssre 10375	A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
Theorem	iccssxr 10376	A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
⊢ (𝐴[,]𝐵) ⊆ ℝ*
Theorem	iocssxr 10377	An open-below, closed-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
⊢ (𝐴(,]𝐵) ⊆ ℝ*
Theorem	icossxr 10378	A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
⊢ (𝐴[,)𝐵) ⊆ ℝ*
Theorem	ioossicc 10379	An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
Theorem	icossicc 10380	A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵)
Theorem	iocssicc 10381	A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
Theorem	ioossico 10382	An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)
Theorem	iocssioo 10383	Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶(,]𝐷) ⊆ (𝐴(,)𝐵))
Theorem	icossioo 10384	Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴(,)𝐵))
Theorem	ioossioo 10385	Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
Theorem	iccsupr 10386*	A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
Theorem	elioopnf 10387	Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
Theorem	elioomnf 10388	Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴)))
Theorem	elicopnf 10389	Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵)))
Theorem	repos 10390	Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
⊢ (𝐴 ∈ (0(,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	ioof 10391	The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
Theorem	iccf 10392	The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
Theorem	unirnioo 10393	The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
⊢ ℝ = ∪ ran (,)
Theorem	dfioo2 10394*	Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤 ∧ 𝑤 < 𝑦)})
Theorem	ioorebasg 10395	Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,))
Theorem	elrege0 10396	The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
Theorem	rge0ssre 10397	Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
⊢ (0[,)+∞) ⊆ ℝ
Theorem	elxrge0 10398	Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴))
Theorem	0e0icopnf 10399	0 is a member of (0[,)+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ∈ (0[,)+∞)
Theorem	0e0iccpnf 10400	0 is a member of (0[,]+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ∈ (0[,]+∞)