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	xltadd1 10101	Extended real version of ltadd1 8599. (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 10102	Extended real version of ltadd2 8589. (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 10103	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 10104	Extended real version of le2add 8614. (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 10105	Extended real version of lt2add 8615. Note that ltleadd 8616, 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 10106	Extended real version of subge0 8645. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
Theorem	xposdif 10107	Extended real version of posdif 8625. (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 10108	Under certain conditions, the conclusion of lesubadd 8604 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 10109	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 10110	Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8338. (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 10111	Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 10110. (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 10112	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 10113	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 10114	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 10115	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 10116	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 10117*	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 10118*	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 10119*	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 10120*	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 10121*	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 10122*	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 10123*	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 10124*	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 10125*	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 10126*	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 10127*	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 10128*	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 10129*	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 10130*	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 10131*	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 10132	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 10133*	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 10134	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 10135	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 10136	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 10137	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 10138	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 10139	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 10140*	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 10141	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 10142	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 10143*	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 10144*	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 10145*	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 10146	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 ) ) )
Theorem	elioc1 10147	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 ) ) )
Theorem	elico1 10148	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 ) ) )
Theorem	elicc1 10149	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 ) ) )
Theorem	iccid 10150	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 } )
Theorem	icc0r 10151	An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( B < A -> ( A [,] B ) = (/) ) )
Theorem	eliooxr 10152	An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
|- ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) )
Theorem	eliooord 10153	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 ) )
Theorem	ubioc1 10154	The upper bound belongs to an open-below, closed-above interval. See ubicc2 10210. (Contributed by FL, 29-May-2014.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) )
Theorem	lbico1 10155	The lower bound belongs to a closed-below, open-above interval. See lbicc2 10209. (Contributed by FL, 29-May-2014.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) )
Theorem	iccleub 10156	An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> C <_ B )
Theorem	iccgelb 10157	An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> A <_ C )
Theorem	elioo5 10158	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 ) ) )
Theorem	elioo4g 10159	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 ) ) )
Theorem	ioossre 10160	An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
|- ( A (,) B ) C_ RR
Theorem	elioc2 10161	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 ) ) )
Theorem	elico2 10162	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 ) ) )
Theorem	elicc2 10163	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 ) ) )
Theorem	elicc2i 10164	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 ) )
Theorem	elicc4 10165	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 ) ) )
Theorem	iccss 10166	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 ) )
Theorem	iccssioo 10167	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 ) )
Theorem	icossico 10168	Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,) D ) C_ ( A [,) B ) )
Theorem	iccss2 10169	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 ) )
Theorem	iccssico 10170	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 ) )
Theorem	iccssioo2 10171	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 ) )
Theorem	iccssico2 10172	Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
|- ( ( C e. ( A [,) B ) /\ D e. ( A [,) B ) ) -> ( C [,] D ) C_ ( A [,) B ) )
Theorem	ioomax 10173	The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
|- ( -oo (,) +oo ) = RR
Theorem	iccmax 10174	The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
|- ( -oo [,] +oo ) = RR*
Theorem	ioopos 10175	The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
|- ( 0 (,) +oo ) = { x e. RR | 0 < x }
Theorem	ioorp 10176	The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( 0 (,) +oo ) = RR+
Theorem	iooshf 10177	Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> A e. ( ( C + B ) (,) ( D + B ) ) ) )
Theorem	iocssre 10178	A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
|- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR )
Theorem	icossre 10179	A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
|- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )
Theorem	iccssre 10180	A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
Theorem	iccssxr 10181	A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
|- ( A [,] B ) C_ RR*
Theorem	iocssxr 10182	An open-below, closed-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
|- ( A (,] B ) C_ RR*
Theorem	icossxr 10183	A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
|- ( A [,) B ) C_ RR*
Theorem	ioossicc 10184	An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
|- ( A (,) B ) C_ ( A [,] B )
Theorem	icossicc 10185	A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
|- ( A [,) B ) C_ ( A [,] B )
Theorem	iocssicc 10186	A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
|- ( A (,] B ) C_ ( A [,] B )
Theorem	ioossico 10187	An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
|- ( A (,) B ) C_ ( A [,) B )
Theorem	iocssioo 10188	Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D < B ) ) -> ( C (,] D ) C_ ( A (,) B ) )
Theorem	icossioo 10189	Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < C /\ D <_ B ) ) -> ( C [,) D ) C_ ( A (,) B ) )
Theorem	ioossioo 10190	Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C (,) D ) C_ ( A (,) B ) )
Theorem	iccsupr 10191*	A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.)
|- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> ( S C_ RR /\ S =/= (/) /\ E. x e. RR A. y e. S y <_ x ) )
Theorem	elioopnf 10192	Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( A e. RR* -> ( B e. ( A (,) +oo ) <-> ( B e. RR /\ A < B ) ) )
Theorem	elioomnf 10193	Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( A e. RR* -> ( B e. ( -oo (,) A ) <-> ( B e. RR /\ B < A ) ) )
Theorem	elicopnf 10194	Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
|- ( A e. RR -> ( B e. ( A [,) +oo ) <-> ( B e. RR /\ A <_ B ) ) )
Theorem	repos 10195	Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
|- ( A e. ( 0 (,) +oo ) <-> ( A e. RR /\ 0 < A ) )
Theorem	ioof 10196	The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- (,) : ( RR* X. RR* ) --> ~P RR
Theorem	iccf 10197	The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- [,] : ( RR* X. RR* ) --> ~P RR*
Theorem	unirnioo 10198	The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
|- RR = U. ran (,)
Theorem	dfioo2 10199*	Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- (,) = ( x e. RR* , y e. RR* |-> { w e. RR | ( x < w /\ w < y ) } )
Theorem	ioorebasg 10200	Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) e. ran (,) )