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Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
4.5.3  Real number intervals
 
Syntaxcioo 9901 Extend class notation with the set of open intervals of extended reals.
 class  (,)
 
Syntaxcioc 9902 Extend class notation with the set of open-below, closed-above intervals of extended reals.
 class  (,]
 
Syntaxcico 9903 Extend class notation with the set of closed-below, open-above intervals of extended reals.
 class  [,)
 
Syntaxcicc 9904 Extend class notation with the set of closed intervals of extended reals.
 class  [,]
 
Definitiondf-ioo 9905* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
 |- 
 (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x  <  z  /\  z  <  y ) } )
 
Definitiondf-ioc 9906* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
 |- 
 (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x  <  z  /\  z  <_  y ) } )
 
Definitiondf-ico 9907* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
 |- 
 [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x  <_  z  /\  z  <  y ) } )
 
Definitiondf-icc 9908* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
 |- 
 [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x  <_  z  /\  z  <_  y ) } )
 
Theoremixxval 9909* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  {
 z  e.  RR*  |  ( A R z  /\  z S B ) }
 )
 
Theoremelixx1 9910* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A O B ) 
 <->  ( C  e.  RR*  /\  A R C  /\  C S B ) ) )
 
Theoremixxf 9911* 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  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  O : (
 RR*  X.  RR* ) --> ~P RR*
 
Theoremixxex 9912* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  O  e.  _V
 
Theoremixxssxr 9913* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  ( A O B )  C_  RR*
 
Theoremelixx3g 9914* 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  e.  RR* and  B  e.  RR*. (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  ( C  e.  ( A O B )  <-> 
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A R C  /\  C S B ) ) )
 
Theoremixxssixx 9915* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   &    |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x T z  /\  z U y ) }
 )   &    |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w 
 ->  A T w ) )   &    |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  ( w S B  ->  w U B ) )   =>    |-  ( A O B )  C_  ( A P B )
 
Theoremixxdisj 9916* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   &    |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x T z  /\  z U y ) }
 )   &    |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B T w  <->  -.  w S B ) )   =>    |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A O B )  i^i  ( B P C ) )  =  (/) )
 
Theoremixxss1 9917* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   &    |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x T z  /\  z S y ) }
 )   &    |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e.  RR* )  ->  (
 ( A W B  /\  B T w ) 
 ->  A R w ) )   =>    |-  ( ( A  e.  RR*  /\  A W B ) 
 ->  ( B P C )  C_  ( A O C ) )
 
Theoremixxss2 9918* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   &    |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z  /\  z T y ) }
 )   &    |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( ( w T B  /\  B W C )  ->  w S C ) )   =>    |-  ( ( C  e.  RR*  /\  B W C )  ->  ( A P B )  C_  ( A O C ) )
 
Theoremixxss12 9919* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   &    |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x T z  /\  z U y ) }
 )   &    |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e.  RR* )  ->  (
 ( A W C  /\  C T w ) 
 ->  A R w ) )   &    |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e.  RR* )  ->  ( ( w U D  /\  D X B )  ->  w S B ) )   =>    |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  ( C P D ) 
 C_  ( A O B ) )
 
Theoremiooex 9920 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |- 
 (,)  e.  _V
 
Theoremiooval 9921* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
 
Theoremiooidg 9922 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
 |-  ( A  e.  RR*  ->  ( A (,) A )  =  (/) )
 
Theoremelioo3g 9923 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  e.  RR* and  B  e.  RR*. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  C  <  B ) ) )
 
Theoremelioo1 9924 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  <  C  /\  C  <  B ) ) )
 
Theoremelioore 9925 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  e.  ( B (,) C )  ->  A  e.  RR )
 
Theoremlbioog 9926 An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  A  e.  ( A (,) B ) )
 
Theoremubioog 9927 An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  B  e.  ( A (,) B ) )
 
Theoremiooval2 9928* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR  |  ( A  <  x  /\  x  <  B ) } )
 
Theoremiooss1 9929 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( A  e.  RR*  /\  A  <_  B )  ->  ( B (,) C )  C_  ( A (,) C ) )
 
Theoremiooss2 9930 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( C  e.  RR*  /\  B  <_  C )  ->  ( A (,) B )  C_  ( A (,) C ) )
 
Theoremiocval 9931* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,] B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) } )
 
Theoremicoval 9932* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  { x  e.  RR*  |  ( A 
 <_  x  /\  x  <  B ) } )
 
Theoremiccval 9933* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,] B )  =  { x  e.  RR*  |  ( A 
 <_  x  /\  x  <_  B ) } )
 
Theoremelioo2 9934 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR  /\  A  <  C  /\  C  <  B ) ) )
 
Theoremelioc1 9935 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  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  <  C  /\  C  <_  B ) ) )
 
Theoremelico1 9936 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  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) ) )
 
Theoremelicc1 9937 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) ) )
 
Theoremiccid 9938 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
 |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A }
 )
 
Theoremicc0r 9939 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  A  ->  ( A [,] B )  =  (/) ) )
 
Theoremeliooxr 9940 An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  ( B (,) C )  ->  ( B  e.  RR*  /\  C  e.  RR* ) )
 
Theoremeliooord 9941 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  e.  ( B (,) C )  ->  ( B  <  A  /\  A  <  C ) )
 
Theoremubioc1 9942 The upper bound belongs to an open-below, closed-above interval. See ubicc2 9998. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  B  e.  ( A (,] B ) )
 
Theoremlbico1 9943 The lower bound belongs to a closed-below, open-above interval. See lbicc2 9997. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
 
Theoremiccleub 9944 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  C  <_  B )
 
Theoremiccgelb 9945 An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  A  <_  C )
 
Theoremelioo5 9946 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C 
 /\  C  <  B ) ) )
 
Theoremelioo4g 9947 Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )
 
Theoremioossre 9948 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
 |-  ( A (,) B )  C_  RR
 
Theoremelioc2 9949 Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  <  C  /\  C  <_  B )
 ) )
 
Theoremelico2 9950 Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <-> 
 ( C  e.  RR  /\  A  <_  C  /\  C  <  B ) ) )
 
Theoremelicc2 9951 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A [,] B )  <-> 
 ( C  e.  RR  /\  A  <_  C  /\  C  <_  B ) ) )
 
Theoremelicc2i 9952 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( C  e.  ( A [,] B )  <->  ( C  e.  RR  /\  A  <_  C  /\  C  <_  B )
 )
 
Theoremelicc4 9953 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( A  <_  C 
 /\  C  <_  B ) ) )
 
Theoremiccss 9954 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  e.  RR  /\  B  e.  RR )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C [,] D ) 
 C_  ( A [,] B ) )
 
Theoremiccssioo 9955 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <  B ) )  ->  ( C [,] D ) 
 C_  ( A (,) B ) )
 
Theoremicossico 9956 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  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <_  B ) )  ->  ( C [,) D )  C_  ( A [,) B ) )
 
Theoremiccss2 9957 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  e.  ( A [,] B ) 
 /\  D  e.  ( A [,] B ) ) 
 ->  ( C [,] D )  C_  ( A [,] B ) )
 
Theoremiccssico 9958 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <  B ) )  ->  ( C [,] D )  C_  ( A [,) B ) )
 
Theoremiccssioo2 9959 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( C  e.  ( A (,) B ) 
 /\  D  e.  ( A (,) B ) ) 
 ->  ( C [,] D )  C_  ( A (,) B ) )
 
Theoremiccssico2 9960 Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( C  e.  ( A [,) B ) 
 /\  D  e.  ( A [,) B ) ) 
 ->  ( C [,] D )  C_  ( A [,) B ) )
 
Theoremioomax 9961 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
 |-  ( -oo (,) +oo )  =  RR
 
Theoremiccmax 9962 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
 |-  ( -oo [,] +oo )  =  RR*
 
Theoremioopos 9963 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
 |-  ( 0 (,) +oo )  =  { x  e.  RR  |  0  < 
 x }
 
Theoremioorp 9964 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( 0 (,) +oo )  =  RR+
 
Theoremiooshf 9965 Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  -  B )  e.  ( C (,) D )  <->  A  e.  (
 ( C  +  B ) (,) ( D  +  B ) ) ) )
 
Theoremiocssre 9966 A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
 
Theoremicossre 9967 A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  ( A [,) B )  C_  RR )
 
Theoremiccssre 9968 A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B )  C_  RR )
 
Theoremiccssxr 9969 A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
 |-  ( A [,] B )  C_  RR*
 
Theoremiocssxr 9970 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 )  C_  RR*
 
Theoremicossxr 9971 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 )  C_  RR*
 
Theoremioossicc 9972 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
 |-  ( A (,) B )  C_  ( A [,] B )
 
Theoremicossicc 9973 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
 |-  ( A [,) B )  C_  ( A [,] B )
 
Theoremiocssicc 9974 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  ( A (,] B )  C_  ( A [,] B )
 
Theoremioossico 9975 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  ( A (,) B )  C_  ( A [,) B )
 
Theoremiocssioo 9976 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <  B ) )  ->  ( C (,] D )  C_  ( A (,) B ) )
 
Theoremicossioo 9977 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <_  B ) )  ->  ( C [,) D ) 
 C_  ( A (,) B ) )
 
Theoremioossioo 9978 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <_  B ) )  ->  ( C (,) D )  C_  ( A (,) B ) )
 
Theoremiccsupr 9979* 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  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  ( S  C_  RR  /\  S  =/=  (/)  /\  E. x  e.  RR  A. y  e.  S  y  <_  x ) )
 
Theoremelioopnf 9980 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  RR*  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  <  B ) ) )
 
Theoremelioomnf 9981 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  RR*  ->  ( B  e.  ( -oo (,) A )  <->  ( B  e.  RR  /\  B  <  A ) ) )
 
Theoremelicopnf 9982 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( A  e.  RR  ->  ( B  e.  ( A [,) +oo )  <->  ( B  e.  RR  /\  A  <_  B ) ) )
 
Theoremrepos 9983 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
 |-  ( A  e.  (
 0 (,) +oo )  <->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremioof 9984 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.)
 |- 
 (,) : ( RR*  X.  RR* )
 --> ~P RR
 
Theoremiccf 9985 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.)
 |- 
 [,] : ( RR*  X.  RR* )
 --> ~P RR*
 
Theoremunirnioo 9986 The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
 |- 
 RR  =  U. ran  (,)
 
Theoremdfioo2 9987* 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  e.  RR* ,  y  e.  RR*  |->  { w  e.  RR  |  ( x  <  w  /\  w  <  y ) } )
 
Theoremioorebasg 9988 Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e.  ran  (,) )
 
Theoremelrege0 9989 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  (
 0 [,) +oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
 
Theoremrge0ssre 9990 Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
 |-  ( 0 [,) +oo )  C_  RR
 
Theoremelxrge0 9991 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( A  e.  (
 0 [,] +oo )  <->  ( A  e.  RR*  /\  0  <_  A ) )
 
Theorem0e0icopnf 9992 0 is a member of  ( 0 [,) +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  e.  ( 0 [,) +oo )
 
Theorem0e0iccpnf 9993 0 is a member of  ( 0 [,] +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  e.  ( 0 [,] +oo )
 
Theoremge0addcl 9994 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ( A  e.  ( 0 [,) +oo )  /\  B  e.  (
 0 [,) +oo ) ) 
 ->  ( A  +  B )  e.  ( 0 [,) +oo ) )
 
Theoremge0mulcl 9995 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ( A  e.  ( 0 [,) +oo )  /\  B  e.  (
 0 [,) +oo ) ) 
 ->  ( A  x.  B )  e.  ( 0 [,) +oo ) )
 
Theoremge0xaddcl 9996 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
 0 [,] +oo ) ) 
 ->  ( A +e B )  e.  (
 0 [,] +oo ) )
 
Theoremlbicc2 9997 The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B ) )
 
Theoremubicc2 9998 The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B ) )
 
Theorem0elunit 9999 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  0  e.  ( 0 [,] 1 )
 
Theorem1elunit 10000 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  1  e.  ( 0 [,] 1 )
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