P. 100 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xrltletr 9901	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 ) )
Theorem	xrletr 9902	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 ) )
Theorem	xrlttrd 9903	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 )
Theorem	xrlelttrd 9904	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 )
Theorem	xrltletrd 9905	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 )
Theorem	xrletrd 9906	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 )
Theorem	xrltne 9907	'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B =/= A )
Theorem	nltpnft 9908	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 ) )
Theorem	npnflt 9909	An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
|- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )
Theorem	xgepnf 9910	An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
|- ( A e. RR* -> ( +oo <_ A <-> A = +oo ) )
Theorem	ngtmnft 9911	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 ) )
Theorem	nmnfgt 9912	An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
|- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )
Theorem	xrrebnd 9913	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 ) ) )
Theorem	xrre 9914	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 )
Theorem	xrre2 9915	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 )
Theorem	xrre3 9916	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 )
Theorem	ge0gtmnf 9917	A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ 0 <_ A ) -> -oo < A )
Theorem	ge0nemnf 9918	A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ 0 <_ A ) -> A =/= -oo )
Theorem	xrrege0 9919	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 )
Theorem	z2ge 9920*	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 ) )
Theorem	xnegeq 9921	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 )
Theorem	xnegpnf 9922	Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
|- -e +oo = -oo
Theorem	xnegmnf 9923	Minus -oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
|- -e -oo = +oo
Theorem	rexneg 9924	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 )
Theorem	xneg0 9925	The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- -e 0 = 0
Theorem	xnegcl 9926	Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e A e. RR* )
Theorem	xnegneg 9927	Extended real version of negneg 8295. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e -e A = A )
Theorem	xneg11 9928	Extended real version of neg11 8296. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( -e A = -e B <-> A = B ) )
Theorem	xltnegi 9929	Forward direction of xltneg 9930. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )
Theorem	xltneg 9930	Extended real version of ltneg 8508. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
Theorem	xleneg 9931	Extended real version of leneg 8511. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -e B <_ -e A ) )
Theorem	xlt0neg1 9932	Extended real version of lt0neg1 8514. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A < 0 <-> 0 < -e A ) )
Theorem	xlt0neg2 9933	Extended real version of lt0neg2 8515. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 < A <-> -e A < 0 ) )
Theorem	xle0neg1 9934	Extended real version of le0neg1 8516. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) )
Theorem	xle0neg2 9935	Extended real version of le0neg2 8517. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( 0 <_ A <-> -e A <_ 0 ) )
Theorem	xrpnfdc 9936	An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = +oo )
Theorem	xrmnfdc 9937	An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = -oo )
Theorem	xaddf 9938	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- +e : ( RR* X. RR* ) --> RR*
Theorem	xaddval 9939	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 ) ) ) ) ) )
Theorem	xaddpnf1 9940	Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
Theorem	xaddpnf2 9941	Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= -oo ) -> ( +oo +e A ) = +oo )
Theorem	xaddmnf1 9942	Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
Theorem	xaddmnf2 9943	Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo )
Theorem	pnfaddmnf 9944	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
Theorem	mnfaddpnf 9945	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
Theorem	rexadd 9946	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 ) )
Theorem	rexsub 9947	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 ) )
Theorem	rexaddd 9948	The extended real addition operation when both arguments are real. Deduction version of rexadd 9946. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A +e B ) = ( A + B ) )
Theorem	xnegcld 9949	Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> -e A e. RR* )
Theorem	xrex 9950	The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
|- RR* e. _V
Theorem	xaddnemnf 9951	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 )
Theorem	xaddnepnf 9952	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 )
Theorem	xnegid 9953	Extended real version of negid 8292. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e -e A ) = 0 )
Theorem	xaddcl 9954	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* )
Theorem	xaddcom 9955	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 ) )
Theorem	xaddid1 9956	Extended real version of addrid 8183. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e 0 ) = A )
Theorem	xaddid2 9957	Extended real version of addlid 8184. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 +e A ) = A )
Theorem	xaddid1d 9958	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 )
Theorem	xnn0lenn0nn0 9959	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 )
Theorem	xnn0le2is012 9960	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 ) )
Theorem	xnn0xadd0 9961	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 ) ) )
Theorem	xnegdi 9962	Extended real version of negdi 8302. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
Theorem	xaddass 9963	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 9964, 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 ) ) )
Theorem	xaddass2 9964	Associativity of extended real addition. See xaddass 9963 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 ) ) )
Theorem	xpncan 9965	Extended real version of pncan 8251. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A )
Theorem	xnpcan 9966	Extended real version of npcan 8254. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
Theorem	xleadd1a 9967	Extended real version of leadd1 8476; 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 ) )
Theorem	xleadd2a 9968	Commuted form of xleadd1a 9967. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( C +e A ) <_ ( C +e B ) )
Theorem	xleadd1 9969	Weakened version of xleadd1a 9967 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 ) ) )
Theorem	xltadd1 9970	Extended real version of ltadd1 8475. (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 ) ) )
Theorem	xltadd2 9971	Extended real version of ltadd2 8465. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( A < B <-> ( C +e A ) < ( C +e B ) ) )
Theorem	xaddge0 9972	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 ) )
Theorem	xle2add 9973	Extended real version of le2add 8490. (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 ) ) )
Theorem	xlt2add 9974	Extended real version of lt2add 8491. Note that ltleadd 8492, 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 ) ) )
Theorem	xsubge0 9975	Extended real version of subge0 8521. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
Theorem	xposdif 9976	Extended real version of posdif 8501. (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 ) ) )
Theorem	xlesubadd 9977	Under certain conditions, the conclusion of lesubadd 8480 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 ) ) )
Theorem	xaddcld 9978	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* )
Theorem	xadd4d 9979	Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8214. (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 ) ) )
Theorem	xnn0add4d 9980	Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 9979. (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 ) ) )
Theorem	xleaddadd 9981	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
Syntax	cioo 9982	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 9983	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 9984	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 9985	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 9986*	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 ) } )
Definition	df-ioc 9987*	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 ) } )
Definition	df-ico 9988*	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 ) } )
Definition	df-icc 9989*	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 ) } )
Theorem	ixxval 9990*	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 ) } )
Theorem	elixx1 9991*	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 ) ) )
Theorem	ixxf 9992*	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*
Theorem	ixxex 9993*	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
Theorem	ixxssxr 9994*	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*
Theorem	elixx3g 9995*	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 ) ) )
Theorem	ixxssixx 9996*	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 )
Theorem	ixxdisj 9997*	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 ) ) = (/) )
Theorem	ixxss1 9998*	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 ) )
Theorem	ixxss2 9999*	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 ) )
Theorem	ixxss12 10000*	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 ) )