Type | Label | Description |
Statement |
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4.5.3 Real number intervals
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Syntax | cioo 9901 |
Extend class notation with the set of open intervals of extended reals.
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Syntax | cioc 9902 |
Extend class notation with the set of open-below, closed-above intervals
of extended reals.
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![(,] (,]](_ioc.gif) |
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Syntax | cico 9903 |
Extend class notation with the set of closed-below, open-above intervals
of extended reals.
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Syntax | cicc 9904 |
Extend class notation with the set of closed intervals of extended
reals.
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![[,] [,]](_icc.gif) |
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Definition | df-ioo 9905* |
Define the set of open intervals of extended reals. (Contributed by NM,
24-Dec-2006.)
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Definition | df-ioc 9906* |
Define the set of open-below, closed-above intervals of extended reals.
(Contributed by NM, 24-Dec-2006.)
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Definition | df-ico 9907* |
Define the set of closed-below, open-above intervals of extended reals.
(Contributed by NM, 24-Dec-2006.)
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Definition | df-icc 9908* |
Define the set of closed intervals of extended reals. (Contributed by
NM, 24-Dec-2006.)
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Theorem | ixxval 9909* |
Value of the interval function. (Contributed by Mario Carneiro,
3-Nov-2013.)
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Theorem | elixx1 9910* |
Membership in an interval of extended reals. (Contributed by Mario
Carneiro, 3-Nov-2013.)
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Theorem | ixxf 9911* |
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.)
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Theorem | ixxex 9912* |
The set of intervals of extended reals exists. (Contributed by Mario
Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
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Theorem | ixxssxr 9913* |
The set of intervals of extended reals maps to subsets of extended
reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
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Theorem | elixx3g 9914* |
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
and .
(Contributed by Mario Carneiro,
3-Nov-2013.)
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Theorem | ixxssixx 9915* |
An interval is a subset of its closure. (Contributed by Paul Chapman,
18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | ixxdisj 9916* |
Split an interval into disjoint pieces. (Contributed by Mario
Carneiro, 16-Jun-2014.)
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Theorem | ixxss1 9917* |
Subset relationship for intervals of extended reals. (Contributed by
Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro,
28-Apr-2015.)
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Theorem | ixxss2 9918* |
Subset relationship for intervals of extended reals. (Contributed by
Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro,
28-Apr-2015.)
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Theorem | ixxss12 9919* |
Subset relationship for intervals of extended reals. (Contributed by
Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro,
28-Apr-2015.)
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Theorem | iooex 9920 |
The set of open intervals of extended reals exists. (Contributed by NM,
6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | iooval 9921* |
Value of the open interval function. (Contributed by NM, 24-Dec-2006.)
(Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | iooidg 9922 |
An open interval with identical lower and upper bounds is empty.
(Contributed by Jim Kingdon, 29-Mar-2020.)
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Theorem | elioo3g 9923 |
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
and .
(Contributed by NM, 24-Dec-2006.)
(Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | elioo1 9924 |
Membership in an open interval of extended reals. (Contributed by NM,
24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | elioore 9925 |
A member of an open interval of reals is a real. (Contributed by NM,
17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | lbioog 9926 |
An open interval does not contain its left endpoint. (Contributed by
Jim Kingdon, 30-Mar-2020.)
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Theorem | ubioog 9927 |
An open interval does not contain its right endpoint. (Contributed by
Jim Kingdon, 30-Mar-2020.)
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Theorem | iooval2 9928* |
Value of the open interval function. (Contributed by NM, 6-Feb-2007.)
(Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | iooss1 9929 |
Subset relationship for open intervals of extended reals. (Contributed
by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
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Theorem | iooss2 9930 |
Subset relationship for open intervals of extended reals. (Contributed
by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | iocval 9931* |
Value of the open-below, closed-above interval function. (Contributed
by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
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     ![(,] (,]](_ioc.gif) 
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Theorem | icoval 9932* |
Value of the closed-below, open-above interval function. (Contributed
by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | iccval 9933* |
Value of the closed interval function. (Contributed by NM,
24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
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     ![[,] [,]](_icc.gif) 
 
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Theorem | elioo2 9934 |
Membership in an open interval of extended reals. (Contributed by NM,
6-Feb-2007.)
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Theorem | elioc1 9935 |
Membership in an open-below, closed-above interval of extended reals.
(Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro,
3-Nov-2013.)
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      ![(,] (,]](_ioc.gif)       |
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Theorem | elico1 9936 |
Membership in a closed-below, open-above interval of extended reals.
(Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro,
3-Nov-2013.)
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Theorem | elicc1 9937 |
Membership in a closed interval of extended reals. (Contributed by NM,
24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
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      ![[,] [,]](_icc.gif)  
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Theorem | iccid 9938 |
A closed interval with identical lower and upper bounds is a singleton.
(Contributed by Jeff Hankins, 13-Jul-2009.)
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   ![[,] [,]](_icc.gif)      |
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Theorem | icc0r 9939 |
An empty closed interval of extended reals. (Contributed by Jim
Kingdon, 30-Mar-2020.)
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      ![[,] [,]](_icc.gif) 
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Theorem | eliooxr 9940 |
An inhabited open interval spans an interval of extended reals.
(Contributed by NM, 17-Aug-2008.)
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Theorem | eliooord 9941 |
Ordering implied by a member of an open interval of reals. (Contributed
by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.)
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Theorem | ubioc1 9942 |
The upper bound belongs to an open-below, closed-above interval. See
ubicc2 9998. (Contributed by FL, 29-May-2014.)
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     ![(,] (,]](_ioc.gif)    |
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Theorem | lbico1 9943 |
The lower bound belongs to a closed-below, open-above interval. See
lbicc2 9997. (Contributed by FL, 29-May-2014.)
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Theorem | iccleub 9944 |
An element of a closed interval is less than or equal to its upper bound.
(Contributed by Jeff Hankins, 14-Jul-2009.)
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    ![[,] [,]](_icc.gif)  
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Theorem | iccgelb 9945 |
An element of a closed interval is more than or equal to its lower bound
(Contributed by Thierry Arnoux, 23-Dec-2016.)
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    ![[,] [,]](_icc.gif)  
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Theorem | elioo5 9946 |
Membership in an open interval of extended reals. (Contributed by NM,
17-Aug-2008.)
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Theorem | elioo4g 9947 |
Membership in an open interval of extended reals. (Contributed by NM,
8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
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Theorem | ioossre 9948 |
An open interval is a set of reals. (Contributed by NM,
31-May-2007.)
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Theorem | elioc2 9949 |
Membership in an open-below, closed-above real interval. (Contributed by
Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
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      ![(,] (,]](_ioc.gif)  
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Theorem | elico2 9950 |
Membership in a closed-below, open-above real interval. (Contributed by
Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
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Theorem | elicc2 9951 |
Membership in a closed real interval. (Contributed by Paul Chapman,
21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
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      ![[,] [,]](_icc.gif)  
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Theorem | elicc2i 9952 |
Inference for membership in a closed interval. (Contributed by Scott
Fenton, 3-Jun-2013.)
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   ![[,] [,]](_icc.gif)  
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Theorem | elicc4 9953 |
Membership in a closed real interval. (Contributed by Stefan O'Rear,
16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
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  ![[,] [,]](_icc.gif)       |
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Theorem | iccss 9954 |
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.)
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         ![[,] [,]](_icc.gif)    ![[,] [,]](_icc.gif)    |
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Theorem | iccssioo 9955 |
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20-Feb-2015.)
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      ![[,] [,]](_icc.gif)        |
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Theorem | icossico 9956 |
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.)
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Theorem | iccss2 9957 |
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.)
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    ![[,] [,]](_icc.gif)    ![[,] [,]](_icc.gif)     ![[,] [,]](_icc.gif)    ![[,] [,]](_icc.gif)    |
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Theorem | iccssico 9958 |
Condition for a closed interval to be a subset of a half-open interval.
(Contributed by Mario Carneiro, 9-Sep-2015.)
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      ![[,] [,]](_icc.gif)        |
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Theorem | iccssioo2 9959 |
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20-Feb-2015.)
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             ![[,] [,]](_icc.gif)        |
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Theorem | iccssico2 9960 |
Condition for a closed interval to be a subset of a closed-below,
open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
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             ![[,] [,]](_icc.gif)        |
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Theorem | ioomax 9961 |
The open interval from minus to plus infinity. (Contributed by NM,
6-Feb-2007.)
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Theorem | iccmax 9962 |
The closed interval from minus to plus infinity. (Contributed by Mario
Carneiro, 4-Jul-2014.)
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Theorem | ioopos 9963 |
The set of positive reals expressed as an open interval. (Contributed by
NM, 7-May-2007.)
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Theorem | ioorp 9964 |
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25-Nov-2007.)
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Theorem | iooshf 9965 |
Shift the arguments of the open interval function. (Contributed by NM,
17-Aug-2008.)
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Theorem | iocssre 9966 |
A closed-above interval with real upper bound is a set of reals.
(Contributed by FL, 29-May-2014.)
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     ![(,] (,]](_ioc.gif)    |
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Theorem | icossre 9967 |
A closed-below interval with real lower bound is a set of reals.
(Contributed by Mario Carneiro, 14-Jun-2014.)
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Theorem | iccssre 9968 |
A closed real interval is a set of reals. (Contributed by FL,
6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
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     ![[,] [,]](_icc.gif) 
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Theorem | iccssxr 9969 |
A closed interval is a set of extended reals. (Contributed by FL,
28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
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  ![[,] [,]](_icc.gif)   |
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Theorem | iocssxr 9970 |
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.)
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  ![(,] (,]](_ioc.gif)   |
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Theorem | icossxr 9971 |
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.)
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Theorem | ioossicc 9972 |
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18-Oct-2007.)
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      ![[,] [,]](_icc.gif)   |
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Theorem | icossicc 9973 |
A closed-below, open-above interval is a subset of its closure.
(Contributed by Thierry Arnoux, 25-Oct-2016.)
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      ![[,] [,]](_icc.gif)   |
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Theorem | iocssicc 9974 |
A closed-above, open-below interval is a subset of its closure.
(Contributed by Thierry Arnoux, 1-Apr-2017.)
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  ![(,] (,]](_ioc.gif)    ![[,] [,]](_icc.gif)   |
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Theorem | ioossico 9975 |
An open interval is a subset of its closure-below. (Contributed by
Thierry Arnoux, 3-Mar-2017.)
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Theorem | iocssioo 9976 |
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29-Mar-2017.)
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      ![(,] (,]](_ioc.gif)        |
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Theorem | icossioo 9977 |
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29-Mar-2017.)
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Theorem | ioossioo 9978 |
Condition for an open interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 26-Sep-2017.)
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Theorem | iccsupr 9979* |
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.)
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      ![[,] [,]](_icc.gif)    
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Theorem | elioopnf 9980 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
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Theorem | elioomnf 9981 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
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Theorem | elicopnf 9982 |
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16-Sep-2014.)
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Theorem | repos 9983 |
Two ways of saying that a real number is positive. (Contributed by NM,
7-May-2007.)
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Theorem | ioof 9984 |
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.)
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Theorem | iccf 9985 |
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.)
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![[,]
[,]](_icc.gif)        |
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Theorem | unirnioo 9986 |
The union of the range of the open interval function. (Contributed by
NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
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Theorem | dfioo2 9987* |
Alternate definition of the set of open intervals of extended reals.
(Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro,
1-Sep-2015.)
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Theorem | ioorebasg 9988 |
Open intervals are elements of the set of all open intervals.
(Contributed by Jim Kingdon, 4-Apr-2020.)
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Theorem | elrege0 9989 |
The predicate "is a nonnegative real". (Contributed by Jeff Madsen,
2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
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Theorem | rge0ssre 9990 |
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
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Theorem | elxrge0 9991 |
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28-Jun-2014.)
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Theorem | 0e0icopnf 9992 |
0 is a member of   
(common case). (Contributed by David
A. Wheeler, 8-Dec-2018.)
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Theorem | 0e0iccpnf 9993 |
0 is a member of   
(common case). (Contributed by David
A. Wheeler, 8-Dec-2018.)
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Theorem | ge0addcl 9994 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19-Jun-2014.)
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Theorem | ge0mulcl 9995 |
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19-Jun-2014.)
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Theorem | ge0xaddcl 9996 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 26-Aug-2015.)
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Theorem | lbicc2 9997 |
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.)
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     ![[,] [,]](_icc.gif)    |
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Theorem | ubicc2 9998 |
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.)
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     ![[,] [,]](_icc.gif)    |
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Theorem | 0elunit 9999 |
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
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  ![[,] [,]](_icc.gif)   |
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Theorem | 1elunit 10000 |
One is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
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  ![[,] [,]](_icc.gif)   |