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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrltnr 8701 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  A )
 
Theoremltpnf 8702 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  < +oo )
 
Theorem0ltpnf 8703 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  < +oo
 
Theoremmnflt 8704 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  -> -oo  <  A )
 
Theoremmnflt0 8705 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- -oo  <  0
 
Theoremmnfltpnf 8706 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |- -oo  < +oo
 
Theoremmnfltxr 8707 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
 |-  ( ( A  e.  RR  \/  A  = +oo )  -> -oo  <  A )
 
Theorempnfnlt 8708 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -. +oo  <  A )
 
Theoremnltmnf 8709 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  < -oo )
 
Theorempnfge 8710 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  e.  RR*  ->  A  <_ +oo )
 
Theorem0lepnf 8711 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  <_ +oo
 
Theoremnn0pnfge0 8712 If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 |-  ( ( N  e.  NN0 
 \/  N  = +oo )  ->  0  <_  N )
 
Theoremmnfle 8713 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  e.  RR*  -> -oo  <_  A )
 
Theoremxrltnsym 8714 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )
 
Theoremxrltnsym2 8715 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  ( A  <  B 
 /\  B  <  A ) )
 
Theoremxrlttr 8716 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <  B  /\  B  <  C ) 
 ->  A  <  C ) )
 
Theoremxrltso 8717 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
 |- 
 <  Or  RR*
 
Theoremxrlttri3 8718 Extended real version of lttri3 7098. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
 
Theoremxrltle 8719 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B )
 )
 
Theoremxrleid 8720 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( A  e.  RR*  ->  A  <_  A )
 
Theoremxrletri3 8721 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) ) )
 
Theoremxrlelttr 8722 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 8723 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 8724 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 8725 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 8726 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 8727 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 8728 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 8729 '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 8730 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 8731 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 8732 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 8733 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 8734 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 8735 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 8736 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 -> -oo  <  A )
 
Theoremge0nemnf 8737 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  A  =/= -oo )
 
Theoremxrrege0 8738 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 8739* 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 8740 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 8741 Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  -e +oo  = -oo
 
Theoremxnegmnf 8742 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 8743 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 8744 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e 0  =  0
 
Theoremxnegcl 8745 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e A  e.  RR* )
 
Theoremxnegneg 8746 Extended real version of negneg 7261. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e  -e A  =  A )
 
Theoremxneg11 8747 Extended real version of neg11 7262. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  -e B  <->  A  =  B )
 )
 
Theoremxltnegi 8748 Forward direction of xltneg 8749. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A )
 
Theoremxltneg 8749 Extended real version of ltneg 7457. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
 
Theoremxleneg 8750 Extended real version of leneg 7460. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -e B  <_  -e A ) )
 
Theoremxlt0neg1 8751 Extended real version of lt0neg1 7463. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
 
Theoremxlt0neg2 8752 Extended real version of lt0neg2 7464. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
 
Theoremxle0neg1 8753 Extended real version of le0neg1 7465. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  -e A ) )
 
Theoremxle0neg2 8754 Extended real version of le0neg2 7466. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  -e A  <_  0 ) )
 
Theoremxnegcld 8755 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  -e A  e.  RR* )
 
Theoremxrex 8756 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
 |-  RR*  e.  _V
 
3.5.3  Real number intervals
 
Syntaxcioo 8757 Extend class notation with the set of open intervals of extended reals.
 class  (,)
 
Syntaxcioc 8758 Extend class notation with the set of open-below, closed-above intervals of extended reals.
 class  (,]
 
Syntaxcico 8759 Extend class notation with the set of closed-below, open-above intervals of extended reals.
 class  [,)
 
Syntaxcicc 8760 Extend class notation with the set of closed intervals of extended reals.
 class  [,]
 
Definitiondf-ioo 8761* 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 8762* 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 8763* 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 8764* 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 8765* 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 8766* 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 8767* 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 8768* 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 8769* 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 8770* 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 8771* 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 8772* 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 8773* 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 8774* 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 8775* 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 8776 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 8777* 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 8778 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
 |-  ( A  e.  RR*  ->  ( A (,) A )  =  (/) )
 
Theoremelioo3g 8779 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 8780 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 8781 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 8782 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 8783 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 8784* 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 8785 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 8786 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 8787* 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 8788* 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 8789* 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 8790 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 8791 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 8792 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 8793 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 8794 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 8795 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 8796 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 8797 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 8798 The upper bound belongs to an open-below, closed-above interval. See ubicc2 8853. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  B  e.  ( A (,] B ) )
 
Theoremlbico1 8799 The lower bound belongs to a closed-below, open-above interval. See lbicc2 8852. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
 
Theoremiccleub 8800 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 )
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