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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
5.5.3  Supremum on the extended reals
 
Theoremxrsupexmnf 10501* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)
 |-  ( E. x  e.  RR*  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  RR*  ( y  <  x  ->  E. z  e.  A  y  <  z ) ) 
 ->  E. x  e.  RR*  ( A. y  e.  ( A  u.  {  -oo } )  -.  x  <  y  /\  A. y  e.  RR*  (
 y  <  x  ->  E. z  e.  ( A  u.  {  -oo } )
 y  <  z )
 ) )
 
Theoremxrinfmexpnf 10502* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)
 |-  ( E. x  e.  RR*  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  A  z  <  y ) ) 
 ->  E. x  e.  RR*  ( A. y  e.  ( A  u.  {  +oo } )  -.  y  <  x  /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  ( A  u.  {  +oo } ) z  < 
 y ) ) )
 
Theoremxrsupsslem 10503* Lemma for xrsupss 10505. (Contributed by NM, 25-Oct-2005.)
 |-  ( ( A  C_  RR*  /\  ( A  C_  RR  \/  +oo  e.  A ) )  ->  E. x  e.  RR*  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  RR*  ( y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
Theoremxrinfmsslem 10504* Lemma for xrinfmss 10506. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  C_  RR*  /\  ( A  C_  RR  \/  -oo  e.  A ) )  ->  E. x  e.  RR*  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  A  z  <  y ) ) )
 
Theoremxrsupss 10505* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)
 |-  ( A  C_  RR*  ->  E. x  e.  RR*  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR*  (
 y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
Theoremxrinfmss 10506* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)
 |-  ( A  C_  RR*  ->  E. x  e.  RR*  ( A. y  e.  A  -.  y  <  x  /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  A  z  <  y
 ) ) )
 
Theoremxrinfmss2 10507* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR*  ->  E. x  e.  RR*  ( A. y  e.  A  -.  x `'  <  y  /\  A. y  e.  RR*  ( y `'  <  x 
 ->  E. z  e.  A  y `'  <  z ) ) )
 
Theoremxrub 10508* By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals. (Contributed by NM, 9-Apr-2006.)
 |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( A. x  e. 
 RR  ( x  <  B  ->  E. y  e.  A  x  <  y )  <->  A. x  e.  RR*  ( x  <  B  ->  E. y  e.  A  x  <  y ) ) )
 
Theoremsupxr 10509* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)
 |-  ( ( ( A 
 C_  RR*  /\  B  e.  RR* )  /\  ( A. x  e.  A  -.  B  <  x  /\  A. x  e.  RR  ( x  <  B  ->  E. y  e.  A  x  <  y
 ) ) )  ->  sup ( A ,  RR* ,  <  )  =  B )
 
Theoremsupxr2 10510* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)
 |-  ( ( ( A 
 C_  RR*  /\  B  e.  RR* )  /\  ( A. x  e.  A  x  <_  B  /\  A. x  e.  RR  ( x  <  B  ->  E. y  e.  A  x  <  y ) ) )  ->  sup ( A ,  RR* ,  <  )  =  B )
 
Theoremsupxrcl 10511 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)
 |-  ( A  C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
 
Theoremsupxrun 10512 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  C_  RR*  /\  B  C_  RR*  /\  sup ( A ,  RR* ,  <  ) 
 <_  sup ( B ,  RR*
 ,  <  ) )  ->  sup ( ( A  u.  B ) , 
 RR* ,  <  )  = 
 sup ( B ,  RR*
 ,  <  ) )
 
Theoreminfmxrcl 10513 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by Mario Carneiro, 16-Mar-2014.)
 |-  ( A  C_  RR*  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
 
Theoremsupxrmnf 10514 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  C_  RR*  ->  sup ( ( A  u.  { 
 -oo } ) ,  RR* ,  <  )  =  sup ( A ,  RR* ,  <  ) )
 
Theoremsupxrpnf 10515 The supremum of a set of extended reals containing plus infnity is plus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( ( A  C_  RR*  /\  +oo  e.  A ) 
 ->  sup ( A ,  RR*
 ,  <  )  =  +oo )
 
Theoremsupxrunb1 10516* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  C_  RR*  ->  (
 A. x  e.  RR  E. y  e.  A  x  <_  y  <->  sup ( A ,  RR*
 ,  <  )  =  +oo ) )
 
Theoremsupxrunb2 10517* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  C_  RR*  ->  (
 A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR*
 ,  <  )  =  +oo ) )
 
Theoremsupxrbnd1 10518* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  C_  RR*  ->  ( E. x  e.  RR  A. y  e.  A  y  <  x  <->  sup ( A ,  RR*
 ,  <  )  <  +oo ) )
 
Theoremsupxrbnd2 10519* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  C_  RR*  ->  ( E. x  e.  RR  A. y  e.  A  y 
 <_  x  <->  sup ( A ,  RR*
 ,  <  )  <  +oo ) )
 
Theoremxrsup0 10520 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)
 |- 
 sup ( (/) ,  RR* ,  <  )  =  -oo
 
Theoremsupxrub 10521 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  C_  RR*  /\  B  e.  A ) 
 ->  B  <_  sup ( A ,  RR* ,  <  )
 )
 
Theoremsupxrlub 10522* 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  C_  RR*  /\  B  e.  RR* )  ->  ( B  <  sup ( A ,  RR* ,  <  )  <->  E. x  e.  A  B  <  x ) )
 
Theoremsupxrleub 10523* 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  C_  RR*  /\  B  e.  RR* )  ->  ( sup ( A ,  RR* ,  <  )  <_  B  <->  A. x  e.  A  x  <_  B ) )
 
Theoremsupxrre 10524* 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  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR*
 ,  <  )  =  sup ( A ,  RR ,  <  ) )
 
Theoremsupxrbnd 10525 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  sup ( A ,  RR* ,  <  )  e.  RR )
 
Theoremsupxrgtmnf 10526 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  -oo  <  sup ( A ,  RR*
 ,  <  ) )
 
Theoremsupxrre1 10527 The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR*
 ,  <  )  e.  RR 
 <-> 
 sup ( A ,  RR*
 ,  <  )  <  +oo ) )
 
Theoremsupxrre2 10528 The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR*
 ,  <  )  e.  RR 
 <-> 
 sup ( A ,  RR*
 ,  <  )  =/=  +oo ) )
 
Theoremsupxrss 10529 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  C_  B  /\  B  C_  RR* )  ->  sup ( A ,  RR*
 ,  <  )  <_  sup ( B ,  RR* ,  <  ) )
 
Theoreminfmxrlb 10530 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.)
 |-  ( ( A  C_  RR*  /\  B  e.  A ) 
 ->  sup ( A ,  RR*
 ,  `'  <  )  <_  B )
 
Theoreminfmxrgelb 10531* 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 Mario Carneiro, 6-Sep-2014.)
 |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  A. x  e.  A  B  <_  x ) )
 
Theoreminfmxrre 10532* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
 )  ->  sup ( A ,  RR* ,  `'  <  )  =  sup ( A ,  RR ,  `'  <  ) )
 
Theoremxrinfm0 10533 The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |- 
 sup ( (/) ,  RR* ,  `'  <  )  =  +oo
 
5.5.4  Real number intervals
 
Syntaxcioo 10534 Extend class notation with the set of open intervals of extended reals.
 class  (,)
 
Syntaxcioc 10535 Extend class notation with the set of open-below, closed-above intervals of extended reals.
 class  (,]
 
Syntaxcico 10536 Extend class notation with the set of closed-below, open-above intervals of extended reals.
 class  [,)
 
Syntaxcicc 10537 Extend class notation with the set of closed intervals of extended reals.
 class  [,]
 
Definitiondf-ioo 10538* 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 10539* 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 10540* 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 10541* 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 10542* 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 10543* 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 10544* 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 10545* 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 10546* 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 10547* 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 10548* 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 10549* 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 ) )  =  (/) )
 
Theoremixxun 10550* Split an interval into two parts. (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 ) )   &    |-  Q  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z U y ) } )   &    |-  (
 ( w  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( w S B  /\  B X C ) 
 ->  w U C ) )   &    |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e.  RR* )  ->  ( ( A W B  /\  B T w )  ->  A R w ) )   =>    |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  ( ( A O B )  u.  ( B P C ) )  =  ( A Q C ) )
 
Theoremixxin 10551* Intersection of two intervals 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*  /\  C  e.  RR*  /\  z  e.  RR* )  ->  ( if ( A  <_  C ,  C ,  A ) R z  <->  ( A R z  /\  C R z ) ) )   &    |-  (
 ( z  e.  RR*  /\  B  e.  RR*  /\  D  e.  RR* )  ->  (
 z S if ( B  <_  D ,  B ,  D )  <->  ( z S B  /\  z S D ) ) )   =>    |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A O B )  i^i  ( C O D ) )  =  ( if ( A 
 <_  C ,  C ,  A ) O if ( B  <_  D ,  B ,  D )
 ) )
 
Theoremixxss1 10552* 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 10553* 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 10554* 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 ) )
 
Theoremixxub 10555* Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   &    |-  (
 ( w  e.  RR*  /\  B  e.  RR* )  ->  ( w  <  B  ->  w S B ) )   &    |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  ( w S B  ->  w  <_  B ) )   &    |-  (
 ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A R w ) )   &    |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A  <_  w ) )   =>    |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
 ( A O B ) ,  RR* ,  <  )  =  B )
 
Theoremixxlb 10556* Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  ( x R z 
 /\  z S y ) } )   &    |-  (
 ( w  e.  RR*  /\  B  e.  RR* )  ->  ( w  <  B  ->  w S B ) )   &    |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  ( w S B  ->  w  <_  B ) )   &    |-  (
 ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A R w ) )   &    |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A  <_  w ) )   =>    |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
 ( A O B ) ,  RR* ,  `'  <  )  =  A )
 
Theoremiooex 10557 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 10558* 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 ) } )
 
Theoremioo0 10559 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,) B )  =  (/)  <->  B  <_  A ) )
 
Theoremioon0 10560 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  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,) B )  =/=  (/)  <->  A  <  B ) )
 
Theoremndmioo 10561 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.)
 |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  (/) )
 
Theoremiooid 10562 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 )  =  (/)
 
Theoremelioo3g 10563 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 ) ) )
 
Theoremelioore 10564 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 )
 
Theoremlbioo 10565 An open interval does not contain its left endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |- 
 -.  A  e.  ( A (,) B )
 
Theoremubioo 10566 An open interval does not contain its right endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |- 
 -.  B  e.  ( A (,) B )
 
Theoremiooval2 10567* 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 ) } )
 
Theoremiooin 10568 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A (,) B )  i^i  ( C (,) D ) )  =  ( if ( A  <_  C ,  C ,  A ) (,) if ( B 
 <_  D ,  B ,  D ) ) )
 
Theoremiooss1 10569 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 10570 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 10571* 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 10572* 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 10573* 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 ) } )
 
Theoremelioo1 10574 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 ) ) )
 
Theoremelioo2 10575 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 10576 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 10577 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 10578 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 10579 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 }
 )
 
Theoremico0 10580 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A [,) B )  =  (/)  <->  B  <_  A ) )
 
Theoremioc0 10581 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A (,] B )  =  (/)  <->  B  <_  A ) )
 
Theoremicc0 10582 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A [,] B )  =  (/)  <->  B  <  A ) )
 
Theoremubioc1 10583 The upper bound belongs to an open-below, closed-above interval. See ubicc2 10631. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  B  e.  ( A (,] B ) )
 
Theoremlbico1 10584 The lower bound belongs to a closed-below, open-above interval. See lbicc2 10630. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
 
Theoremiccleub 10585 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeffrey Hankins, 14-Jul-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  C  <_  B )
 
Theoremelioo5 10586 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 ) ) )
 
Theoremeliooxr 10587 A non-empty 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 10588 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 ) )
 
Theoremelioo4g 10589 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 10590 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
 |-  ( A (,) B )  C_  RR
 
Theoremelioc2 10591 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 10592 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 10593 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 10594 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 10595 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 10596 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 10597 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 ) )
 
Theoremiccss2 10598 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 10599 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 10600 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 ) )
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