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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsupxrunb2 13301* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
(𝐴 βŠ† ℝ* β†’ (βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ 𝐴 π‘₯ < 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))
 
Theoremsupxrbnd1 13302* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
(𝐴 βŠ† ℝ* β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 < π‘₯ ↔ sup(𝐴, ℝ*, < ) < +∞))
 
Theoremsupxrbnd2 13303* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
(𝐴 βŠ† ℝ* β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯ ↔ sup(𝐴, ℝ*, < ) < +∞))
 
Theoremxrsup0 13304 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)
sup(βˆ…, ℝ*, < ) = -∞
 
Theoremsupxrub 13305 A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by NM, 7-Feb-2006.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ 𝐴) β†’ 𝐡 ≀ sup(𝐴, ℝ*, < ))
 
Theoremsupxrlub 13306* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by Mario Carneiro, 13-Sep-2015.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐡 < sup(𝐴, ℝ*, < ) ↔ βˆƒπ‘₯ ∈ 𝐴 𝐡 < π‘₯))
 
Theoremsupxrleub 13307* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by NM, 22-Feb-2006.) (Revised by Mario Carneiro, 6-Sep-2014.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (sup(𝐴, ℝ*, < ) ≀ 𝐡 ↔ βˆ€π‘₯ ∈ 𝐴 π‘₯ ≀ 𝐡))
 
Theoremsupxrre 13308* The real and extended real suprema match when the real supremum exists. (Contributed by NM, 18-Oct-2005.) (Proof shortened by Mario Carneiro, 7-Sep-2014.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯) β†’ sup(𝐴, ℝ*, < ) = sup(𝐴, ℝ, < ))
 
Theoremsupxrbnd 13309 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ sup(𝐴, ℝ*, < ) < +∞) β†’ sup(𝐴, ℝ*, < ) ∈ ℝ)
 
Theoremsupxrgtmnf 13310 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ -∞ < sup(𝐴, ℝ*, < ))
 
Theoremsupxrre1 13311 The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) < +∞))
 
Theoremsupxrre2 13312 The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) β‰  +∞))
 
Theoremsupxrss 13313 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 βŠ† 𝐡 ∧ 𝐡 βŠ† ℝ*) β†’ sup(𝐴, ℝ*, < ) ≀ sup(𝐡, ℝ*, < ))
 
Theoreminfxrcl 13314 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by AV, 5-Sep-2020.)
(𝐴 βŠ† ℝ* β†’ inf(𝐴, ℝ*, < ) ∈ ℝ*)
 
Theoreminfxrlb 13315 A member of a set of extended reals is greater than or equal to the set's infimum. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ 𝐴) β†’ inf(𝐴, ℝ*, < ) ≀ 𝐡)
 
Theoreminfxrgelb 13316* The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐡 ≀ inf(𝐴, ℝ*, < ) ↔ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ π‘₯))
 
Theoreminfxrre 13317* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 π‘₯ ≀ 𝑦) β†’ inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ, < ))
 
Theoreminfxrmnf 13318 The infinimum of a set of extended reals containing minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
((𝐴 βŠ† ℝ* ∧ -∞ ∈ 𝐴) β†’ inf(𝐴, ℝ*, < ) = -∞)
 
Theoremxrinf0 13319 The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 5-Sep-2020.)
inf(βˆ…, ℝ*, < ) = +∞
 
Theoreminfxrss 13320 Larger sets of extended reals have smaller infima. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
((𝐴 βŠ† 𝐡 ∧ 𝐡 βŠ† ℝ*) β†’ inf(𝐡, ℝ*, < ) ≀ inf(𝐴, ℝ*, < ))
 
Theoremreltre 13321* For all real numbers there is a smaller real number. (Contributed by AV, 5-Sep-2020.)
βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑦 < π‘₯
 
Theoremrpltrp 13322* For all positive real numbers there is a smaller positive real number. (Contributed by AV, 5-Sep-2020.)
βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ 𝑦 < π‘₯
 
Theoremreltxrnmnf 13323* For all extended real numbers not being minus infinity there is a smaller real number. (Contributed by AV, 5-Sep-2020.)
βˆ€π‘₯ ∈ ℝ* (-∞ < π‘₯ β†’ βˆƒπ‘¦ ∈ ℝ 𝑦 < π‘₯)
 
Theoreminfmremnf 13324 The infimum of the reals is minus infinity. (Contributed by AV, 5-Sep-2020.)
inf(ℝ, ℝ*, < ) = -∞
 
Theoreminfmrp1 13325 The infimum of the positive reals is 0. (Contributed by AV, 5-Sep-2020.)
inf(ℝ+, ℝ, < ) = 0
 
5.5.4  Real number intervals
 
Syntaxcioo 13326 Extend class notation with the set of open intervals of extended reals.
class (,)
 
Syntaxcioc 13327 Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
 
Syntaxcico 13328 Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
 
Syntaxcicc 13329 Extend class notation with the set of closed intervals of extended reals.
class [,]
 
Definitiondf-ioo 13330* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,) = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ < 𝑧 ∧ 𝑧 < 𝑦)})
 
Definitiondf-ioc 13331* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,] = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ < 𝑧 ∧ 𝑧 ≀ 𝑦)})
 
Definitiondf-ico 13332* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,) = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ ≀ 𝑧 ∧ 𝑧 < 𝑦)})
 
Definitiondf-icc 13333* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,] = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ ≀ 𝑧 ∧ 𝑧 ≀ 𝑦)})
 
Theoremixxval 13334* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    β‡’   ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴𝑂𝐡) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐡)})
 
Theoremelixx1 13335* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    β‡’   ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴𝑂𝐡) ↔ (𝐢 ∈ ℝ* ∧ 𝐴𝑅𝐢 ∧ 𝐢𝑆𝐡)))
 
Theoremixxf 13336* 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 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    β‡’   π‘‚:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
 
Theoremixxex 13337* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    β‡’   π‘‚ ∈ V
 
Theoremixxssxr 13338* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    β‡’   (𝐴𝑂𝐡) βŠ† ℝ*
 
Theoremelixx3g 13339* 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 𝐡 ∈ ℝ*. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    β‡’   (𝐢 ∈ (𝐴𝑂𝐡) ↔ ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ (𝐴𝑅𝐢 ∧ 𝐢𝑆𝐡)))
 
Theoremixxssixx 13340* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   π‘ƒ = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑇𝑧 ∧ π‘§π‘ˆπ‘¦)})    &   ((𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ (𝐴𝑅𝑀 β†’ 𝐴𝑇𝑀))    &   ((𝑀 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝑀𝑆𝐡 β†’ π‘€π‘ˆπ΅))    β‡’   (𝐴𝑂𝐡) βŠ† (𝐴𝑃𝐡)
 
Theoremixxdisj 13341* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   π‘ƒ = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑇𝑧 ∧ π‘§π‘ˆπ‘¦)})    &   ((𝐡 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ (𝐡𝑇𝑀 ↔ Β¬ 𝑀𝑆𝐡))    β‡’   ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ ((𝐴𝑂𝐡) ∩ (𝐡𝑃𝐢)) = βˆ…)
 
Theoremixxun 13342* Split an interval into two parts. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   π‘ƒ = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑇𝑧 ∧ π‘§π‘ˆπ‘¦)})    &   ((𝐡 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ (𝐡𝑇𝑀 ↔ Β¬ 𝑀𝑆𝐡))    &   π‘„ = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ π‘§π‘ˆπ‘¦)})    &   ((𝑀 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ ((𝑀𝑆𝐡 ∧ 𝐡𝑋𝐢) β†’ π‘€π‘ˆπΆ))    &   ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ ((π΄π‘Šπ΅ ∧ 𝐡𝑇𝑀) β†’ 𝐴𝑅𝑀))    β‡’   (((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ (π΄π‘Šπ΅ ∧ 𝐡𝑋𝐢)) β†’ ((𝐴𝑂𝐡) βˆͺ (𝐡𝑃𝐢)) = (𝐴𝑄𝐢))
 
Theoremixxin 13343* Intersection of two intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ((𝐴 ∈ ℝ* ∧ 𝐢 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) β†’ (if(𝐴 ≀ 𝐢, 𝐢, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧 ∧ 𝐢𝑅𝑧)))    &   ((𝑧 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) β†’ (𝑧𝑆if(𝐡 ≀ 𝐷, 𝐡, 𝐷) ↔ (𝑧𝑆𝐡 ∧ 𝑧𝑆𝐷)))    β‡’   (((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) β†’ ((𝐴𝑂𝐡) ∩ (𝐢𝑂𝐷)) = (if(𝐴 ≀ 𝐢, 𝐢, 𝐴)𝑂if(𝐡 ≀ 𝐷, 𝐡, 𝐷)))
 
Theoremixxss1 13344* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   π‘ƒ = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑇𝑧 ∧ 𝑧𝑆𝑦)})    &   ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ ((π΄π‘Šπ΅ ∧ 𝐡𝑇𝑀) β†’ 𝐴𝑅𝑀))    β‡’   ((𝐴 ∈ ℝ* ∧ π΄π‘Šπ΅) β†’ (𝐡𝑃𝐢) βŠ† (𝐴𝑂𝐢))
 
Theoremixxss2 13345* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   π‘ƒ = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑇𝑦)})    &   ((𝑀 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ ((𝑀𝑇𝐡 ∧ π΅π‘ŠπΆ) β†’ 𝑀𝑆𝐢))    β‡’   ((𝐢 ∈ ℝ* ∧ π΅π‘ŠπΆ) β†’ (𝐴𝑃𝐡) βŠ† (𝐴𝑂𝐢))
 
Theoremixxss12 13346* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   π‘ƒ = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑇𝑧 ∧ π‘§π‘ˆπ‘¦)})    &   ((𝐴 ∈ ℝ* ∧ 𝐢 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ ((π΄π‘ŠπΆ ∧ 𝐢𝑇𝑀) β†’ 𝐴𝑅𝑀))    &   ((𝑀 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((π‘€π‘ˆπ· ∧ 𝐷𝑋𝐡) β†’ 𝑀𝑆𝐡))    β‡’   (((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (π΄π‘ŠπΆ ∧ 𝐷𝑋𝐡)) β†’ (𝐢𝑃𝐷) βŠ† (𝐴𝑂𝐡))
 
Theoremixxub 13347* Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ((𝑀 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝑀 < 𝐡 β†’ 𝑀𝑆𝐡))    &   ((𝑀 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝑀𝑆𝐡 β†’ 𝑀 ≀ 𝐡))    &   ((𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ (𝐴 < 𝑀 β†’ 𝐴𝑅𝑀))    &   ((𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ (𝐴𝑅𝑀 β†’ 𝐴 ≀ 𝑀))    β‡’   ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ (𝐴𝑂𝐡) β‰  βˆ…) β†’ sup((𝐴𝑂𝐡), ℝ*, < ) = 𝐡)
 
Theoremixxlb 13348* Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by AV, 12-Sep-2020.)
𝑂 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ((𝑀 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝑀 < 𝐡 β†’ 𝑀𝑆𝐡))    &   ((𝑀 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝑀𝑆𝐡 β†’ 𝑀 ≀ 𝐡))    &   ((𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ (𝐴 < 𝑀 β†’ 𝐴𝑅𝑀))    &   ((𝐴 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) β†’ (𝐴𝑅𝑀 β†’ 𝐴 ≀ 𝑀))    β‡’   ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ (𝐴𝑂𝐡) β‰  βˆ…) β†’ inf((𝐴𝑂𝐡), ℝ*, < ) = 𝐴)
 
Theoremiooex 13349 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(,) ∈ V
 
Theoremiooval 13350* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴(,)𝐡) = {π‘₯ ∈ ℝ* ∣ (𝐴 < π‘₯ ∧ π‘₯ < 𝐡)})
 
Theoremioo0 13351 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((𝐴(,)𝐡) = βˆ… ↔ 𝐡 ≀ 𝐴))
 
Theoremioon0 13352 An open interval of extended reals is nonempty iff the lower argument is less than the upper argument. (Contributed by NM, 2-Mar-2007.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((𝐴(,)𝐡) β‰  βˆ… ↔ 𝐴 < 𝐡))
 
Theoremndmioo 13353 The open interval function's value is empty outside of its domain. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
(Β¬ (𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴(,)𝐡) = βˆ…)
 
Theoremiooid 13354 An open interval with identical lower and upper bounds is empty. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐴(,)𝐴) = βˆ…
 
Theoremelioo3g 13355 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 𝐡 ∈ ℝ*. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐢 ∈ (𝐴(,)𝐡) ↔ ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ (𝐴 < 𝐢 ∧ 𝐢 < 𝐡)))
 
Theoremelioore 13356 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 ∈ (𝐡(,)𝐢) β†’ 𝐴 ∈ ℝ)
 
Theoremlbioo 13357 An open interval does not contain its left endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)
¬ 𝐴 ∈ (𝐴(,)𝐡)
 
Theoremubioo 13358 An open interval does not contain its right endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)
¬ 𝐡 ∈ (𝐴(,)𝐡)
 
Theoremiooval2 13359* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴(,)𝐡) = {π‘₯ ∈ ℝ ∣ (𝐴 < π‘₯ ∧ π‘₯ < 𝐡)})
 
Theoremiooin 13360 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) β†’ ((𝐴(,)𝐡) ∩ (𝐢(,)𝐷)) = (if(𝐴 ≀ 𝐢, 𝐢, 𝐴)(,)if(𝐡 ≀ 𝐷, 𝐡, 𝐷)))
 
Theoremiooss1 13361 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
((𝐴 ∈ ℝ* ∧ 𝐴 ≀ 𝐡) β†’ (𝐡(,)𝐢) βŠ† (𝐴(,)𝐢))
 
Theoremiooss2 13362 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐢 ∈ ℝ* ∧ 𝐡 ≀ 𝐢) β†’ (𝐴(,)𝐡) βŠ† (𝐴(,)𝐢))
 
Theoremiocval 13363* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴(,]𝐡) = {π‘₯ ∈ ℝ* ∣ (𝐴 < π‘₯ ∧ π‘₯ ≀ 𝐡)})
 
Theoremicoval 13364* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴[,)𝐡) = {π‘₯ ∈ ℝ* ∣ (𝐴 ≀ π‘₯ ∧ π‘₯ < 𝐡)})
 
Theoremiccval 13365* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴[,]𝐡) = {π‘₯ ∈ ℝ* ∣ (𝐴 ≀ π‘₯ ∧ π‘₯ ≀ 𝐡)})
 
Theoremelioo1 13366 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴(,)𝐡) ↔ (𝐢 ∈ ℝ* ∧ 𝐴 < 𝐢 ∧ 𝐢 < 𝐡)))
 
Theoremelioo2 13367 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴(,)𝐡) ↔ (𝐢 ∈ ℝ ∧ 𝐴 < 𝐢 ∧ 𝐢 < 𝐡)))
 
Theoremelioc1 13368 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* ∧ 𝐡 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴(,]𝐡) ↔ (𝐢 ∈ ℝ* ∧ 𝐴 < 𝐢 ∧ 𝐢 ≀ 𝐡)))
 
Theoremelico1 13369 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* ∧ 𝐡 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴[,)𝐡) ↔ (𝐢 ∈ ℝ* ∧ 𝐴 ≀ 𝐢 ∧ 𝐢 < 𝐡)))
 
Theoremelicc1 13370 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴[,]𝐡) ↔ (𝐢 ∈ ℝ* ∧ 𝐴 ≀ 𝐢 ∧ 𝐢 ≀ 𝐡)))
 
Theoremiccid 13371 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
(𝐴 ∈ ℝ* β†’ (𝐴[,]𝐴) = {𝐴})
 
Theoremico0 13372 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((𝐴[,)𝐡) = βˆ… ↔ 𝐡 ≀ 𝐴))
 
Theoremioc0 13373 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((𝐴(,]𝐡) = βˆ… ↔ 𝐡 ≀ 𝐴))
 
Theoremicc0 13374 An empty closed interval of extended reals. (Contributed by FL, 30-May-2014.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((𝐴[,]𝐡) = βˆ… ↔ 𝐡 < 𝐴))
 
Theoremdfrp2 13375 Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
ℝ+ = (0(,)+∞)
 
Theoremelicod 13376 Membership in a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 ≀ 𝐢)    &   (πœ‘ β†’ 𝐢 < 𝐡)    β‡’   (πœ‘ β†’ 𝐢 ∈ (𝐴[,)𝐡))
 
Theoremicogelb 13377 An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴[,)𝐡)) β†’ 𝐴 ≀ 𝐢)
 
Theoremelicore 13378 A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐢 ∈ (𝐴[,)𝐡)) β†’ 𝐢 ∈ ℝ)
 
Theoremubioc1 13379 The upper bound belongs to an open-below, closed-above interval. See ubicc2 13444. (Contributed by FL, 29-May-2014.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 < 𝐡) β†’ 𝐡 ∈ (𝐴(,]𝐡))
 
Theoremlbico1 13380 The lower bound belongs to a closed-below, open-above interval. See lbicc2 13443. (Contributed by FL, 29-May-2014.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐴 < 𝐡) β†’ 𝐴 ∈ (𝐴[,)𝐡))
 
Theoremiccleub 13381 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴[,]𝐡)) β†’ 𝐢 ≀ 𝐡)
 
Theoremiccgelb 13382 An element of a closed interval is more than or equal to its lower bound. (Contributed by Thierry Arnoux, 23-Dec-2016.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ (𝐴[,]𝐡)) β†’ 𝐴 ≀ 𝐢)
 
Theoremelioo5 13383 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴(,)𝐡) ↔ (𝐴 < 𝐢 ∧ 𝐢 < 𝐡)))
 
Theoremeliooxr 13384 A nonempty open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
(𝐴 ∈ (𝐡(,)𝐢) β†’ (𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*))
 
Theoremeliooord 13385 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 ∈ (𝐡(,)𝐢) β†’ (𝐡 < 𝐴 ∧ 𝐴 < 𝐢))
 
Theoremelioo4g 13386 Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐢 ∈ (𝐴(,)𝐡) ↔ ((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ) ∧ (𝐴 < 𝐢 ∧ 𝐢 < 𝐡)))
 
Theoremioossre 13387 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
(𝐴(,)𝐡) βŠ† ℝ
 
Theoremioosscn 13388 An open interval is a set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐡) βŠ† β„‚
 
Theoremelioc2 13389 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* ∧ 𝐡 ∈ ℝ) β†’ (𝐢 ∈ (𝐴(,]𝐡) ↔ (𝐢 ∈ ℝ ∧ 𝐴 < 𝐢 ∧ 𝐢 ≀ 𝐡)))
 
Theoremelico2 13390 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 ∧ 𝐡 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴[,)𝐡) ↔ (𝐢 ∈ ℝ ∧ 𝐴 ≀ 𝐢 ∧ 𝐢 < 𝐡)))
 
Theoremelicc2 13391 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐢 ∈ (𝐴[,]𝐡) ↔ (𝐢 ∈ ℝ ∧ 𝐴 ≀ 𝐢 ∧ 𝐢 ≀ 𝐡)))
 
Theoremelicc2i 13392 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
𝐴 ∈ ℝ    &   π΅ ∈ ℝ    β‡’   (𝐢 ∈ (𝐴[,]𝐡) ↔ (𝐢 ∈ ℝ ∧ 𝐴 ≀ 𝐢 ∧ 𝐢 ≀ 𝐡))
 
Theoremelicc4 13393 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴[,]𝐡) ↔ (𝐴 ≀ 𝐢 ∧ 𝐢 ≀ 𝐡)))
 
Theoremiccss 13394 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 ∧ 𝐡 ∈ ℝ) ∧ (𝐴 ≀ 𝐢 ∧ 𝐷 ≀ 𝐡)) β†’ (𝐢[,]𝐷) βŠ† (𝐴[,]𝐡))
 
Theoremiccssioo 13395 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (𝐴 < 𝐢 ∧ 𝐷 < 𝐡)) β†’ (𝐢[,]𝐷) βŠ† (𝐴(,)𝐡))
 
Theoremicossico 13396 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* ∧ 𝐡 ∈ ℝ*) ∧ (𝐴 ≀ 𝐢 ∧ 𝐷 ≀ 𝐡)) β†’ (𝐢[,)𝐷) βŠ† (𝐴[,)𝐡))
 
Theoremiccss2 13397 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.)
((𝐢 ∈ (𝐴[,]𝐡) ∧ 𝐷 ∈ (𝐴[,]𝐡)) β†’ (𝐢[,]𝐷) βŠ† (𝐴[,]𝐡))
 
Theoremiccssico 13398 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (𝐴 ≀ 𝐢 ∧ 𝐷 < 𝐡)) β†’ (𝐢[,]𝐷) βŠ† (𝐴[,)𝐡))
 
Theoremiccssioo2 13399 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
((𝐢 ∈ (𝐴(,)𝐡) ∧ 𝐷 ∈ (𝐴(,)𝐡)) β†’ (𝐢[,]𝐷) βŠ† (𝐴(,)𝐡))
 
Theoremiccssico2 13400 Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
((𝐢 ∈ (𝐴[,)𝐡) ∧ 𝐷 ∈ (𝐴[,)𝐡)) β†’ (𝐢[,]𝐷) βŠ† (𝐴[,)𝐡))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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