P. 104 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ixxex 10301*	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 10302*	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 10303*	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 10304*	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 10305*	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 10306*	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 10307*	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 10308*	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 10309	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 10310*	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 10311	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 10312	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 10313	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 10314	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 10315	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 10316	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 10317*	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 10318	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 10319	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 10320*	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 10321*	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 10322*	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 10323	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 10324	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 10325	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 10326	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 10327	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 10328	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 10329	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 10330	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 10331	The upper bound belongs to an open-below, closed-above interval. See ubicc2 10387. (Contributed by FL, 29-May-2014.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) )
Theorem	lbico1 10332	The lower bound belongs to a closed-below, open-above interval. See lbicc2 10386. (Contributed by FL, 29-May-2014.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) )
Theorem	iccleub 10333	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 10334	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 10335	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 10336	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 10337	An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
|- ( A (,) B ) C_ RR
Theorem	elioc2 10338	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 10339	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 10340	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 10341	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 10342	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 10343	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 10344	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 10345	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 10346	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 10347	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 10348	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 10349	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 10350	The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
|- ( -oo (,) +oo ) = RR
Theorem	iccmax 10351	The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
|- ( -oo [,] +oo ) = RR*
Theorem	ioopos 10352	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 10353	The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( 0 (,) +oo ) = RR+
Theorem	iooshf 10354	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 10355	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 10356	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 10357	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 10358	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 10359	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 10360	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 10361	An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
|- ( A (,) B ) C_ ( A [,] B )
Theorem	icossicc 10362	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 10363	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 10364	An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
|- ( A (,) B ) C_ ( A [,) B )
Theorem	iocssioo 10365	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 10366	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 10367	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 10368*	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 10369	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 10370	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 10371	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 10372	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 10373	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 10374	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 10375	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 10376*	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 10377	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 (,) )
Theorem	elrege0 10378	The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
|- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) )
Theorem	rge0ssre 10379	Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
|- ( 0 [,) +oo ) C_ RR
Theorem	elxrge0 10380	Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( A e. ( 0 [,] +oo ) <-> ( A e. RR* /\ 0 <_ A ) )
Theorem	0e0icopnf 10381	0 is a member of ( 0 [,) +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 e. ( 0 [,) +oo )
Theorem	0e0iccpnf 10382	0 is a member of ( 0 [,] +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 e. ( 0 [,] +oo )
Theorem	ge0addcl 10383	The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) -> ( A + B ) e. ( 0 [,) +oo ) )
Theorem	ge0mulcl 10384	The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) -> ( A x. B ) e. ( 0 [,) +oo ) )
Theorem	ge0xaddcl 10385	The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) ) -> ( A +e B ) e. ( 0 [,] +oo ) )
Theorem	lbicc2 10386	The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
Theorem	ubicc2 10387	The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
Theorem	0elunit 10388	Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- 0 e. ( 0 [,] 1 )
Theorem	1elunit 10389	One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- 1 e. ( 0 [,] 1 )
Theorem	iooneg 10390	Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A (,) B ) <-> -u C e. ( -u B (,) -u A ) ) )
Theorem	iccneg 10391	Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( A [,] B ) <-> -u C e. ( -u B [,] -u A ) ) )
Theorem	icoshft 10392	A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) -> ( X + C ) e. ( ( A + C ) [,) ( B + C ) ) ) )
Theorem	icoshftf1o 10393*	Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
|- F = ( x e. ( A [,) B ) |-> ( x + C ) )   =>    |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) )
Theorem	icodisj 10394	End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A [,) B ) i^i ( B [,) C ) ) = (/) )
Theorem	ioodisj 10395	If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = (/) )
Theorem	iccshftr 10396	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A + R ) = C   &    |- ( B + R ) = D   =>    |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X + R ) e. ( C [,] D ) ) )
Theorem	iccshftri 10397	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. RR   &    |- B e. RR   &    |- R e. RR   &    |- ( A + R ) = C   &    |- ( B + R ) = D   =>    |- ( X e. ( A [,] B ) -> ( X + R ) e. ( C [,] D ) )
Theorem	iccshftl 10398	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A - R ) = C   &    |- ( B - R ) = D   =>    |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
Theorem	iccshftli 10399	Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. RR   &    |- B e. RR   &    |- R e. RR   &    |- ( A - R ) = C   &    |- ( B - R ) = D   =>    |- ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) )
Theorem	iccdil 10400	Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A x. R ) = C   &    |- ( B x. R ) = D   =>    |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) )