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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  e.  RR*  ->  A  <_  A )
 
Theoremxrletri3 9202 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) ) )
 
Theoremxrlelttr 9203 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <_  B  /\  B  <  C ) 
 ->  A  <  C ) )
 
Theoremxrltletr 9204 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <  B  /\  B  <_  C )  ->  A  <  C ) )
 
Theoremxrletr 9205 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <_  B  /\  B  <_  C )  ->  A  <_  C )
 )
 
Theoremxrlttrd 9206 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremxrlelttrd 9207 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremxrltletrd 9208 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B  <_  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremxrletrd 9209 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <_  C )
 
Theoremxrltne 9210 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  B  =/=  A )
 
Theoremnltpnft 9211 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
 
Theoremngtmnft 9212 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
 |-  ( A  e.  RR*  ->  ( A  = -oo  <->  -. -oo 
 <  A ) )
 
Theoremxrrebnd 9213 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
 |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
 
Theoremxrre 9214 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B ) )  ->  A  e.  RR )
 
Theoremxrre2 9215 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR )
 
Theoremxrre3 9216 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B 
 <_  A  /\  A  < +oo ) )  ->  A  e.  RR )
 
Theoremge0gtmnf 9217 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 -> -oo  <  A )
 
Theoremge0nemnf 9218 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  A  =/= -oo )
 
Theoremxrrege0 9219 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( 0 
 <_  A  /\  A  <_  B ) )  ->  A  e.  RR )
 
Theoremz2ge 9220* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  E. k  e.  ZZ  ( M  <_  k  /\  N  <_  k ) )
 
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  -> 
 -e A  =  -e B )
 
Theoremxnegpnf 9222 Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  -e +oo  = -oo
 
Theoremxnegmnf 9223 Minus -oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  -e -oo  = +oo
 
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  e.  RR  -> 
 -e A  =  -u A )
 
Theoremxneg0 9225 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e 0  =  0
 
Theoremxnegcl 9226 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e A  e.  RR* )
 
Theoremxnegneg 9227 Extended real version of negneg 7676. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e  -e A  =  A )
 
Theoremxneg11 9228 Extended real version of neg11 7677. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  -e B  <->  A  =  B )
 )
 
Theoremxltnegi 9229 Forward direction of xltneg 9230. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A )
 
Theoremxltneg 9230 Extended real version of ltneg 7884. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
 
Theoremxleneg 9231 Extended real version of leneg 7887. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -e B  <_  -e A ) )
 
Theoremxlt0neg1 9232 Extended real version of lt0neg1 7890. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
 
Theoremxlt0neg2 9233 Extended real version of lt0neg2 7891. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
 
Theoremxle0neg1 9234 Extended real version of le0neg1 7892. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  -e A ) )
 
Theoremxle0neg2 9235 Extended real version of le0neg2 7893. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  -e A  <_  0 ) )
 
Theoremxnegcld 9236 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  -e A  e.  RR* )
 
Theoremxrex 9237 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
 |-  RR*  e.  _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  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x  <  z  /\  z  <  y ) } )
 
Definitiondf-ioc 9243* 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 9244* 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 9245* 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 9246* 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 9247* 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 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  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  O : (
 RR*  X.  RR* ) --> ~P RR*
 
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  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  O  e.  _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  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   =>    |-  ( A O B )  C_  RR*
 
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  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 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  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 9253* 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 9254* 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 9255* 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 9256* 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 9257 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 9258* 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 9259 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
 |-  ( A  e.  RR*  ->  ( 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  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 9261 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 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  e.  ( B (,) C )  ->  A  e.  RR )
 
Theoremlbioog 9263 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 9264 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 9265* 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 9266 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 9267 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 9268* 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 9269* 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 9270* 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 9271 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 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  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  <  C  /\  C  <_  B ) ) )
 
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  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  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  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) ) )
 
Theoremiccid 9275 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 9276 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 9277 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 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  e.  ( B (,) C )  ->  ( B  <  A  /\  A  <  C ) )
 
Theoremubioc1 9279 The upper bound belongs to an open-below, closed-above interval. See ubicc2 9334. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  B  e.  ( 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  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  A  e.  ( 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  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  C  <_  B )
 
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  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  A  <_  C )
 
Theoremelioo5 9283 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 9284 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 9285 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
 |-  ( A (,) B )  C_  RR
 
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  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  <  C  /\  C  <_  B )
 ) )
 
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  e.  RR  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <-> 
 ( C  e.  RR  /\  A  <_  C  /\  C  <  B ) ) )
 
Theoremelicc2 9288 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 9289 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 9290 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 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  e.  RR  /\  B  e.  RR )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C [,] D ) 
 C_  ( 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  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <  B ) )  ->  ( C [,] D ) 
 C_  ( 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  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <_  B ) )  ->  ( C [,) D )  C_  ( 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  e.  ( A [,] B ) 
 /\  D  e.  ( A [,] B ) ) 
 ->  ( C [,] D )  C_  ( 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  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <  B ) )  ->  ( C [,] D )  C_  ( 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  e.  ( A (,) B ) 
 /\  D  e.  ( A (,) B ) ) 
 ->  ( C [,] D )  C_  ( 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  e.  ( A [,) B ) 
 /\  D  e.  ( A [,) B ) ) 
 ->  ( C [,] D )  C_  ( A [,) B ) )
 
Theoremioomax 9298 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
 |-  ( -oo (,) +oo )  =  RR
 
Theoremiccmax 9299 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
 |-  ( -oo [,] +oo )  =  RR*
 
Theoremioopos 9300 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
 |-  ( 0 (,) +oo )  =  { x  e.  RR  |  0  < 
 x }
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