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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrleid 9201 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
(𝐴 ∈ ℝ*𝐴𝐴)
 
Theoremxrletri3 9202 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremxrlelttr 9203 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Theoremxrltletr 9204 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 < 𝐵𝐵𝐶) → 𝐴 < 𝐶))
 
Theoremxrletr 9205 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremxrlttrd 9206 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑𝐴 < 𝐶)
 
Theoremxrlelttrd 9207 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑𝐴 < 𝐶)
 
Theoremxrltletrd 9208 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴 < 𝐶)
 
Theoremxrletrd 9209 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremxrltne 9210 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐵𝐴)
 
Theoremnltpnft 9211 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
 
Theoremngtmnft 9212 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
 
Theoremxrrebnd 9213 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴𝐴 < +∞)))
 
Theoremxrre 9214 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (-∞ < 𝐴𝐴𝐵)) → 𝐴 ∈ ℝ)
 
Theoremxrre2 9215 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵 < 𝐶)) → 𝐵 ∈ ℝ)
 
Theoremxrre3 9216 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (𝐵𝐴𝐴 < +∞)) → 𝐴 ∈ ℝ)
 
Theoremge0gtmnf 9217 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)
 
Theoremge0nemnf 9218 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
 
Theoremxrrege0 9219 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴𝐴𝐵)) → 𝐴 ∈ ℝ)
 
Theoremz2ge 9220* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∃𝑘 ∈ ℤ (𝑀𝑘𝑁𝑘))
 
Theoremxnegeq 9221 Equality of two extended numbers with -𝑒 in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
(𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
 
Theoremxnegpnf 9222 Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
-𝑒+∞ = -∞
 
Theoremxnegmnf 9223 Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
-𝑒-∞ = +∞
 
Theoremrexneg 9224 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴)
 
Theoremxneg0 9225 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒0 = 0
 
Theoremxnegcl 9226 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)
 
Theoremxnegneg 9227 Extended real version of negneg 7676. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)
 
Theoremxneg11 9228 Extended real version of neg11 7677. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵𝐴 = 𝐵))
 
Theoremxltnegi 9229 Forward direction of xltneg 9230. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)
 
Theoremxltneg 9230 Extended real version of ltneg 7884. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))
 
Theoremxleneg 9231 Extended real version of leneg 7887. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))
 
Theoremxlt0neg1 9232 Extended real version of lt0neg1 7890. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
 
Theoremxlt0neg2 9233 Extended real version of lt0neg2 7891. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
 
Theoremxle0neg1 9234 Extended real version of le0neg1 7892. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))
 
Theoremxle0neg2 9235 Extended real version of le0neg2 7893. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))
 
Theoremxnegcld 9236 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -𝑒𝐴 ∈ ℝ*)
 
Theoremxrex 9237 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
* ∈ V
 
3.5.3  Real number intervals
 
Syntaxcioo 9238 Extend class notation with the set of open intervals of extended reals.
class (,)
 
Syntaxcioc 9239 Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
 
Syntaxcico 9240 Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
 
Syntaxcicc 9241 Extend class notation with the set of closed intervals of extended reals.
class [,]
 
Definitiondf-ioo 9242* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
 
Definitiondf-ioc 9243* 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 < 𝑧𝑧𝑦)})
 
Definitiondf-ico 9244* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
 
Definitiondf-icc 9245* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
 
Theoremixxval 9246* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
 
Theoremelixx1 9247* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝑅𝐶𝐶𝑆𝐵)))
 
Theoremixxf 9248* 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
 
Theoremixxex 9249* 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       𝑂 ∈ V
 
Theoremixxssxr 9250* 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       (𝐴𝑂𝐵) ⊆ ℝ*
 
Theoremelixx3g 9251* 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝑅𝐶𝐶𝑆𝐵)))
 
Theoremixxssixx 9252* 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴𝑅𝑤𝐴𝑇𝑤))    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝑈𝐵))       (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)
 
Theoremixxdisj 9253* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) = ∅)
 
Theoremixxss1 9254* 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑆𝑦)})    &   ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵𝐵𝑇𝑤) → 𝐴𝑅𝑤))       ((𝐴 ∈ ℝ*𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))
 
Theoremixxss2 9255* 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑇𝑦)})    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑤𝑇𝐵𝐵𝑊𝐶) → 𝑤𝑆𝐶))       ((𝐶 ∈ ℝ*𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))
 
Theoremixxss12 9256* 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* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝑊𝐶𝐶𝑇𝑤) → 𝐴𝑅𝑤))    &   ((𝑤 ∈ ℝ*𝐷 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤𝑈𝐷𝐷𝑋𝐵) → 𝑤𝑆𝐵))       (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝑊𝐶𝐷𝑋𝐵)) → (𝐶𝑃𝐷) ⊆ (𝐴𝑂𝐵))
 
Theoremiooex 9257 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(,) ∈ V
 
Theoremiooval 9258* 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 < 𝑥𝑥 < 𝐵)})
 
Theoremiooidg 9259 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
(𝐴 ∈ ℝ* → (𝐴(,)𝐴) = ∅)
 
Theoremelioo3g 9260 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 < 𝐶𝐶 < 𝐵)))
 
Theoremelioo1 9261 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 < 𝐶𝐶 < 𝐵)))
 
Theoremelioore 9262 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)
 
Theoremlbioog 9263 An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ¬ 𝐴 ∈ (𝐴(,)𝐵))
 
Theoremubioog 9264 An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴(,)𝐵))
 
Theoremiooval2 9265* 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 < 𝑥𝑥 < 𝐵)})
 
Theoremiooss1 9266 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
((𝐴 ∈ ℝ*𝐴𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
 
Theoremiooss2 9267 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐶 ∈ ℝ*𝐵𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
 
Theoremiocval 9268* 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 < 𝑥𝑥𝐵)})
 
Theoremicoval 9269* 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* ∣ (𝐴𝑥𝑥 < 𝐵)})
 
Theoremiccval 9270* 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* ∣ (𝐴𝑥𝑥𝐵)})
 
Theoremelioo2 9271 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶𝐶 < 𝐵)))
 
Theoremelioc1 9272 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 < 𝐶𝐶𝐵)))
 
Theoremelico1 9273 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*𝐴𝐶𝐶 < 𝐵)))
 
Theoremelicc1 9274 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*𝐴𝐶𝐶𝐵)))
 
Theoremiccid 9275 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
(𝐴 ∈ ℝ* → (𝐴[,]𝐴) = {𝐴})
 
Theoremicc0r 9276 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → (𝐴[,]𝐵) = ∅))
 
Theoremeliooxr 9277 An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
(𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ*𝐶 ∈ ℝ*))
 
Theoremeliooord 9278 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 < 𝐴𝐴 < 𝐶))
 
Theoremubioc1 9279 The upper bound belongs to an open-below, closed-above interval. See ubicc2 9334. (Contributed by FL, 29-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵))
 
Theoremlbico1 9280 The lower bound belongs to a closed-below, open-above interval. See lbicc2 9333. (Contributed by FL, 29-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
 
Theoremiccleub 9281 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)) → 𝐶𝐵)
 
Theoremiccgelb 9282 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)) → 𝐴𝐶)
 
Theoremelioo5 9283 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶𝐶 < 𝐵)))
 
Theoremelioo4g 9284 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 < 𝐶𝐶 < 𝐵)))
 
Theoremioossre 9285 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
(𝐴(,)𝐵) ⊆ ℝ
 
Theoremelioc2 9286 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 < 𝐶𝐶𝐵)))
 
Theoremelico2 9287 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 ∧ 𝐴𝐶𝐶 < 𝐵)))
 
Theoremelicc2 9288 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 ∧ 𝐴𝐶𝐶𝐵)))
 
Theoremelicc2i 9289 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴𝐶𝐶𝐵))
 
Theoremelicc4 9290 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) ↔ (𝐴𝐶𝐶𝐵)))
 
Theoremiccss 9291 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) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
 
Theoremiccssioo 9292 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 < 𝐶𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
 
Theoremicossico 9293 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*) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴[,)𝐵))
 
Theoremiccss2 9294 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))
 
Theoremiccssico 9295 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*) ∧ (𝐴𝐶𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))
 
Theoremiccssioo2 9296 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))
 
Theoremiccssico2 9297 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))
 
Theoremioomax 9298 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
(-∞(,)+∞) = ℝ
 
Theoremiccmax 9299 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
(-∞[,]+∞) = ℝ*
 
Theoremioopos 9300 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
(0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
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