P. 102 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xrre3 10101	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 10102	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 10103	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 10104	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 10105*	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 10106	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 10107	Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
|- -e +oo = -oo
Theorem	xnegmnf 10108	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 10109	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 10110	The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- -e 0 = 0
Theorem	xnegcl 10111	Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e A e. RR* )
Theorem	xnegneg 10112	Extended real version of negneg 8471. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e -e A = A )
Theorem	xneg11 10113	Extended real version of neg11 8472. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( -e A = -e B <-> A = B ) )
Theorem	xltnegi 10114	Forward direction of xltneg 10115. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )
Theorem	xltneg 10115	Extended real version of ltneg 8684. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
Theorem	xleneg 10116	Extended real version of leneg 8687. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -e B <_ -e A ) )
Theorem	xlt0neg1 10117	Extended real version of lt0neg1 8690. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A < 0 <-> 0 < -e A ) )
Theorem	xlt0neg2 10118	Extended real version of lt0neg2 8691. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 < A <-> -e A < 0 ) )
Theorem	xle0neg1 10119	Extended real version of le0neg1 8692. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) )
Theorem	xle0neg2 10120	Extended real version of le0neg2 8693. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( 0 <_ A <-> -e A <_ 0 ) )
Theorem	xrpnfdc 10121	An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = +oo )
Theorem	xrmnfdc 10122	An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = -oo )
Theorem	xaddf 10123	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- +e : ( RR* X. RR* ) --> RR*
Theorem	xaddval 10124	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 10125	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 10126	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 10127	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 10128	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 10129	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 10130	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 10131	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 10132	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 10133	The extended real addition operation when both arguments are real. Deduction version of rexadd 10131. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A +e B ) = ( A + B ) )
Theorem	xnegcld 10134	Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> -e A e. RR* )
Theorem	xrex 10135	The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
|- RR* e. _V
Theorem	xaddnemnf 10136	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 10137	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 10138	Extended real version of negid 8468. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e -e A ) = 0 )
Theorem	xaddcl 10139	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 10140	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 10141	Extended real version of addrid 8359. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e 0 ) = A )
Theorem	xaddid2 10142	Extended real version of addlid 8360. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 +e A ) = A )
Theorem	xaddid1d 10143	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 10144	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 10145	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 10146	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 10147	Extended real version of negdi 8478. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
Theorem	xaddass 10148	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 10149, 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 10149	Associativity of extended real addition. See xaddass 10148 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 10150	Extended real version of pncan 8427. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A )
Theorem	xnpcan 10151	Extended real version of npcan 8430. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
Theorem	xleadd1a 10152	Extended real version of leadd1 8652; 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 10153	Commuted form of xleadd1a 10152. (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 10154	Weakened version of xleadd1a 10152 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 10155	Extended real version of ltadd1 8651. (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 10156	Extended real version of ltadd2 8641. (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 10157	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 10158	Extended real version of le2add 8666. (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 10159	Extended real version of lt2add 8667. Note that ltleadd 8668, 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 10160	Extended real version of subge0 8697. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
Theorem	xposdif 10161	Extended real version of posdif 8677. (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 10162	Under certain conditions, the conclusion of lesubadd 8656 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 10163	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 10164	Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8390. (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 10165	Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 10164. (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 10166	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 10167	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 10168	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 10169	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 10170	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 10171*	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 10172*	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 10173*	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 10174*	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 10175*	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 10176*	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 10177*	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 10178*	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 10179*	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 10180*	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 10181*	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 10182*	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 10183*	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 10184*	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 10185*	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 ) )
Theorem	iooex 10186	The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- (,) e. _V
Theorem	iooval 10187*	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 ) } )
Theorem	iooidg 10188	An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
|- ( A e. RR* -> ( A (,) A ) = (/) )
Theorem	elioo3g 10189	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 ) ) )
Theorem	elioo1 10190	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 ) ) )
Theorem	elioore 10191	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 )
Theorem	lbioog 10192	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 ) )
Theorem	ubioog 10193	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 ) )
Theorem	iooval2 10194*	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 ) } )
Theorem	iooss1 10195	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 ) )
Theorem	iooss2 10196	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 ) )
Theorem	iocval 10197*	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 ) } )
Theorem	icoval 10198*	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 ) } )
Theorem	iccval 10199*	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 ) } )
Theorem	elioo2 10200	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 ) ) )