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Theorem List for Intuitionistic Logic Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxnegeq 9801 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 9802 Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  -e +oo  = -oo
 
Theoremxnegmnf 9803 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 9804 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 9805 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e 0  =  0
 
Theoremxnegcl 9806 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e A  e.  RR* )
 
Theoremxnegneg 9807 Extended real version of negneg 8184. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e  -e A  =  A )
 
Theoremxneg11 9808 Extended real version of neg11 8185. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  -e B  <->  A  =  B )
 )
 
Theoremxltnegi 9809 Forward direction of xltneg 9810. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A )
 
Theoremxltneg 9810 Extended real version of ltneg 8396. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
 
Theoremxleneg 9811 Extended real version of leneg 8399. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -e B  <_  -e A ) )
 
Theoremxlt0neg1 9812 Extended real version of lt0neg1 8402. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
 
Theoremxlt0neg2 9813 Extended real version of lt0neg2 8403. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
 
Theoremxle0neg1 9814 Extended real version of le0neg1 8404. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  -e A ) )
 
Theoremxle0neg2 9815 Extended real version of le0neg2 8405. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  -e A  <_  0 ) )
 
Theoremxrpnfdc 9816 An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
 |-  ( A  e.  RR*  -> DECID  A  = +oo )
 
Theoremxrmnfdc 9817 An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
 |-  ( A  e.  RR*  -> DECID  A  = -oo )
 
Theoremxaddf 9818 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 +e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxaddval 9819 Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) ) )
 
Theoremxaddpnf1 9820 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
 
Theoremxaddpnf2 9821 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
 
Theoremxaddmnf1 9822 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
 
Theoremxaddmnf2 9823 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
 
Theorempnfaddmnf 9824 Addition of positive and negative infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( +oo +e -oo )  =  0
 
Theoremmnfaddpnf 9825 Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( -oo +e +oo )  =  0
 
Theoremrexadd 9826 The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e B )  =  ( A  +  B ) )
 
Theoremrexsub 9827 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e  -e B )  =  ( A  -  B ) )
 
Theoremrexaddd 9828 The extended real addition operation when both arguments are real. Deduction version of rexadd 9826. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A +e B )  =  ( A  +  B ) )
 
Theoremxnegcld 9829 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  -e A  e.  RR* )
 
Theoremxrex 9830 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
 |-  RR*  e.  _V
 
Theoremxaddnemnf 9831 Closure of extended real addition in the subset  RR*  /  { -oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo ) )  ->  ( A +e B )  =/= -oo )
 
Theoremxaddnepnf 9832 Closure of extended real addition in the subset  RR*  /  { +oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo ) )  ->  ( A +e B )  =/= +oo )
 
Theoremxnegid 9833 Extended real version of negid 8181. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e  -e A )  =  0 )
 
Theoremxaddcl 9834 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
 
Theoremxaddcom 9835 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e A ) )
 
Theoremxaddid1 9836 Extended real version of addid1 8072. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
 
Theoremxaddid2 9837 Extended real version of addid2 8073. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 +e A )  =  A )
 
Theoremxaddid1d 9838  0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  ( A +e 0 )  =  A )
 
Theoremxnn0lenn0nn0 9839 An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.)
 |-  ( ( M  e. NN0*  /\  N  e.  NN0  /\  M  <_  N )  ->  M  e.  NN0 )
 
Theoremxnn0le2is012 9840 An extended nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 24-Nov-2021.)
 |-  ( ( N  e. NN0*  /\  N  <_  2 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
 
Theoremxnn0xadd0 9841 The sum of two extended nonnegative integers is  0 iff each of the two extended nonnegative integers is 
0. (Contributed by AV, 14-Dec-2020.)
 |-  ( ( A  e. NN0*  /\  B  e. NN0* )  ->  ( ( A +e B )  =  0  <-> 
 ( A  =  0 
 /\  B  =  0 ) ) )
 
Theoremxnegdi 9842 Extended real version of negdi 8191. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> 
 -e ( A +e B )  =  (  -e A +e  -e B ) )
 
Theoremxaddass 9843 Associativity of extended real addition. The correct condition here is "it is not the case that both +oo and -oo appear as one of  A ,  B ,  C, i.e.  -.  { +oo , -oo }  C_  { A ,  B ,  C }", but this condition is difficult to work with, so we break the theorem into two parts: this one, where -oo is not present in  A ,  B ,  C, and xaddass2 9844, where +oo is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  ->  (
 ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
 
Theoremxaddass2 9844 Associativity of extended real addition. See xaddass 9843 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )  /\  ( C  e.  RR*  /\  C  =/= +oo ) )  ->  (
 ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
 
Theoremxpncan 9845 Extended real version of pncan 8140. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A +e B ) +e  -e B )  =  A )
 
Theoremxnpcan 9846 Extended real version of npcan 8143. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A +e  -e B ) +e B )  =  A )
 
Theoremxleadd1a 9847 Extended real version of leadd1 8364; note that the converse implication is not true, unlike the real version (for example  0  <  1 but  ( 1 +e +oo )  <_  ( 0 +e +oo )). (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( A +e C )  <_  ( B +e C ) )
 
Theoremxleadd2a 9848 Commuted form of xleadd1a 9847. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( C +e A )  <_  ( C +e B ) )
 
Theoremxleadd1 9849 Weakened version of xleadd1a 9847 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( A +e C )  <_  ( B +e C ) ) )
 
Theoremxltadd1 9850 Extended real version of ltadd1 8363. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
 
Theoremxltadd2 9851 Extended real version of ltadd2 8353. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C +e A )  <  ( C +e B ) ) )
 
Theoremxaddge0 9852 The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0 
 <_  A  /\  0  <_  B ) )  -> 
 0  <_  ( A +e B ) )
 
Theoremxle2add 9853 Extended real version of le2add 8378. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A  <_  C  /\  B  <_  D )  ->  ( A +e B )  <_  ( C +e D ) ) )
 
Theoremxlt2add 9854 Extended real version of lt2add 8379. Note that ltleadd 8380, which has weaker assumptions, is not true for the extended reals (since  0  + +oo  <  1  + +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A  <  C  /\  B  <  D ) 
 ->  ( A +e B )  <  ( C +e D ) ) )
 
Theoremxsubge0 9855 Extended real version of subge0 8409. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  ( A +e  -e B )  <->  B  <_  A ) )
 
Theoremxposdif 9856 Extended real version of posdif 8389. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
 
Theoremxlesubadd 9857 Under certain conditions, the conclusion of lesubadd 8368 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( 0  <_  A  /\  B  =/= -oo  /\  0  <_  C ) ) 
 ->  ( ( A +e  -e B ) 
 <_  C  <->  A  <_  ( C +e B ) ) )
 
Theoremxaddcld 9858 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   =>    |-  ( ph  ->  ( A +e B )  e.  RR* )
 
Theoremxadd4d 9859 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8103. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  ( ph  ->  ( A  e.  RR*  /\  A  =/= -oo ) )   &    |-  ( ph  ->  ( B  e.  RR*  /\  B  =/= -oo )
 )   &    |-  ( ph  ->  ( C  e.  RR*  /\  C  =/= -oo ) )   &    |-  ( ph  ->  ( D  e.  RR*  /\  D  =/= -oo )
 )   =>    |-  ( ph  ->  (
 ( A +e B ) +e
 ( C +e D ) )  =  ( ( A +e C ) +e
 ( B +e D ) ) )
 
Theoremxnn0add4d 9860 Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 9859. (Contributed by AV, 12-Dec-2020.)
 |-  ( ph  ->  A  e. NN0* )   &    |-  ( ph  ->  B  e. NN0* )   &    |-  ( ph  ->  C  e. NN0* )   &    |-  ( ph  ->  D  e. NN0* )   =>    |-  ( ph  ->  (
 ( A +e B ) +e
 ( C +e D ) )  =  ( ( A +e C ) +e
 ( B +e D ) ) )
 
Theoremxleaddadd 9861 Cancelling a factor of two in  <_ (expressed as addition rather than as a factor to avoid extended real multiplication). (Contributed by Jim Kingdon, 18-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  ( A +e A )  <_  ( B +e B ) ) )
 
4.5.3  Real number intervals
 
Syntaxcioo 9862 Extend class notation with the set of open intervals of extended reals.
 class  (,)
 
Syntaxcioc 9863 Extend class notation with the set of open-below, closed-above intervals of extended reals.
 class  (,]
 
Syntaxcico 9864 Extend class notation with the set of closed-below, open-above intervals of extended reals.
 class  [,)
 
Syntaxcicc 9865 Extend class notation with the set of closed intervals of extended reals.
 class  [,]
 
Definitiondf-ioo 9866* 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 9867* 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 9868* 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 9869* 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 9870* 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 9871* 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 9872* 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 9873* 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 9874* 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 9875* 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 9876* 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 9877* 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 9878* 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 9879* 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 9880* 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 9881 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 9882* 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 9883 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
 |-  ( A  e.  RR*  ->  ( A (,) A )  =  (/) )
 
Theoremelioo3g 9884 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 9885 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 9886 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 9887 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 9888 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 9889* 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 9890 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 9891 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 9892* 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 9893* 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 9894* 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 9895 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 9896 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 9897 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 9898 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 9899 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 9900 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  A  ->  ( A [,] B )  =  (/) ) )
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