Type  Label  Description 
Statement 

Theorem  iccgelb 8801 
An element of a closed interval is more than or equal to its lower bound
(Contributed by Thierry Arnoux, 23Dec2016.)



Theorem  elioo5 8802 
Membership in an open interval of extended reals. (Contributed by NM,
17Aug2008.)



Theorem  elioo4g 8803 
Membership in an open interval of extended reals. (Contributed by NM,
8Jun2007.) (Revised by Mario Carneiro, 28Apr2015.)



Theorem  ioossre 8804 
An open interval is a set of reals. (Contributed by NM,
31May2007.)



Theorem  elioc2 8805 
Membership in an openbelow, closedabove real interval. (Contributed
by Paul Chapman, 30Dec2007.) (Revised by Mario Carneiro,
14Jun2014.)



Theorem  elico2 8806 
Membership in a closedbelow, openabove real interval. (Contributed by
Paul Chapman, 21Jan2008.) (Revised by Mario Carneiro,
14Jun2014.)



Theorem  elicc2 8807 
Membership in a closed real interval. (Contributed by Paul Chapman,
21Sep2007.) (Revised by Mario Carneiro, 14Jun2014.)



Theorem  elicc2i 8808 
Inference for membership in a closed interval. (Contributed by Scott
Fenton, 3Jun2013.)



Theorem  elicc4 8809 
Membership in a closed real interval. (Contributed by Stefan O'Rear,
16Nov2014.) (Proof shortened by Mario Carneiro, 1Jan2017.)



Theorem  iccss 8810 
Condition for a closed interval to be a subset of another closed
interval. (Contributed by Jeff Madsen, 2Sep2009.) (Revised by Mario
Carneiro, 20Feb2015.)



Theorem  iccssioo 8811 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20Feb2015.)



Theorem  icossico 8812 
Condition for a closedbelow, openabove interval to be a subset of a
closedbelow, openabove interval. (Contributed by Thierry Arnoux,
21Sep2017.)



Theorem  iccss2 8813 
Condition for a closed interval to be a subset of another closed
interval. (Contributed by Jeff Madsen, 2Sep2009.) (Revised by Mario
Carneiro, 28Apr2015.)



Theorem  iccssico 8814 
Condition for a closed interval to be a subset of a halfopen interval.
(Contributed by Mario Carneiro, 9Sep2015.)



Theorem  iccssioo2 8815 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20Feb2015.)



Theorem  iccssico2 8816 
Condition for a closed interval to be a subset of a closedbelow,
openabove interval. (Contributed by Mario Carneiro, 20Feb2015.)



Theorem  ioomax 8817 
The open interval from minus to plus infinity. (Contributed by NM,
6Feb2007.)



Theorem  iccmax 8818 
The closed interval from minus to plus infinity. (Contributed by Mario
Carneiro, 4Jul2014.)



Theorem  ioopos 8819 
The set of positive reals expressed as an open interval. (Contributed by
NM, 7May2007.)



Theorem  ioorp 8820 
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25Nov2007.)



Theorem  iooshf 8821 
Shift the arguments of the open interval function. (Contributed by NM,
17Aug2008.)



Theorem  iocssre 8822 
A closedabove interval with real upper bound is a set of reals.
(Contributed by FL, 29May2014.)



Theorem  icossre 8823 
A closedbelow interval with real lower bound is a set of reals.
(Contributed by Mario Carneiro, 14Jun2014.)



Theorem  iccssre 8824 
A closed real interval is a set of reals. (Contributed by FL,
6Jun2007.) (Proof shortened by Paul Chapman, 21Jan2008.)



Theorem  iccssxr 8825 
A closed interval is a set of extended reals. (Contributed by FL,
28Jul2008.) (Revised by Mario Carneiro, 4Jul2014.)



Theorem  iocssxr 8826 
An openbelow, closedabove interval is a subset of the extended reals.
(Contributed by FL, 29May2014.) (Revised by Mario Carneiro,
4Jul2014.)



Theorem  icossxr 8827 
A closedbelow, openabove interval is a subset of the extended reals.
(Contributed by FL, 29May2014.) (Revised by Mario Carneiro,
4Jul2014.)



Theorem  ioossicc 8828 
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18Oct2007.)



Theorem  icossicc 8829 
A closedbelow, openabove interval is a subset of its closure.
(Contributed by Thierry Arnoux, 25Oct2016.)



Theorem  iocssicc 8830 
A closedabove, openbelow interval is a subset of its closure.
(Contributed by Thierry Arnoux, 1Apr2017.)



Theorem  ioossico 8831 
An open interval is a subset of its closurebelow. (Contributed by
Thierry Arnoux, 3Mar2017.)



Theorem  iocssioo 8832 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29Mar2017.)



Theorem  icossioo 8833 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29Mar2017.)



Theorem  ioossioo 8834 
Condition for an open interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 26Sep2017.)



Theorem  iccsupr 8835* 
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, 21Jan2008.)



Theorem  elioopnf 8836 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)



Theorem  elioomnf 8837 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)



Theorem  elicopnf 8838 
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16Sep2014.)



Theorem  repos 8839 
Two ways of saying that a real number is positive. (Contributed by NM,
7May2007.)



Theorem  ioof 8840 
The set of open intervals of extended reals maps to subsets of reals.
(Contributed by NM, 7Feb2007.) (Revised by Mario Carneiro,
16Nov2013.)



Theorem  iccf 8841 
The set of closed intervals of extended reals maps to subsets of
extended reals. (Contributed by FL, 14Jun2007.) (Revised by Mario
Carneiro, 3Nov2013.)



Theorem  unirnioo 8842 
The union of the range of the open interval function. (Contributed by
NM, 7May2007.) (Revised by Mario Carneiro, 30Jan2014.)



Theorem  dfioo2 8843* 
Alternate definition of the set of open intervals of extended reals.
(Contributed by NM, 1Mar2007.) (Revised by Mario Carneiro,
1Sep2015.)



Theorem  ioorebasg 8844 
Open intervals are elements of the set of all open intervals.
(Contributed by Jim Kingdon, 4Apr2020.)



Theorem  elrege0 8845 
The predicate "is a nonnegative real". (Contributed by Jeff Madsen,
2Sep2009.) (Proof shortened by Mario Carneiro, 18Jun2014.)



Theorem  rge0ssre 8846 
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9Sep2018.) (Proof shortened by AV, 8Sep2019.)



Theorem  elxrge0 8847 
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28Jun2014.)



Theorem  0e0icopnf 8848 
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8Dec2018.)



Theorem  0e0iccpnf 8849 
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8Dec2018.)



Theorem  ge0addcl 8850 
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19Jun2014.)



Theorem  ge0mulcl 8851 
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19Jun2014.)



Theorem  lbicc2 8852 
The lower bound of a closed interval is a member of it. (Contributed by
Paul Chapman, 26Nov2007.) (Revised by FL, 29May2014.) (Revised by
Mario Carneiro, 9Sep2015.)



Theorem  ubicc2 8853 
The upper bound of a closed interval is a member of it. (Contributed by
Paul Chapman, 26Nov2007.) (Revised by FL, 29May2014.)



Theorem  0elunit 8854 
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)



Theorem  1elunit 8855 
One is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)



Theorem  iooneg 8856 
Membership in a negated open real interval. (Contributed by Paul Chapman,
26Nov2007.)



Theorem  iccneg 8857 
Membership in a negated closed real interval. (Contributed by Paul
Chapman, 26Nov2007.)



Theorem  icoshft 8858 
A shifted real is a member of a shifted, closedbelow, openabove real
interval. (Contributed by Paul Chapman, 25Mar2008.)



Theorem  icoshftf1o 8859* 
Shifting a closedbelow, openabove interval is onetoone onto.
(Contributed by Paul Chapman, 25Mar2008.) (Proof shortened by Mario
Carneiro, 1Sep2015.)



Theorem  icodisj 8860 
Endtoend closedbelow, openabove real intervals are disjoint.
(Contributed by Mario Carneiro, 16Jun2014.)



Theorem  ioodisj 8861 
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, 13Jul2009.)



Theorem  iccshftr 8862 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftri 8863 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftl 8864 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftli 8865 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccdil 8866 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccdili 8867 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  icccntr 8868 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  icccntri 8869 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  divelunit 8870 
A condition for a ratio to be a member of the closed unit. (Contributed
by Scott Fenton, 11Jun2013.)



Theorem  lincmb01cmp 8871 
A linear combination of two reals which lies in the interval between
them. (Contributed by Jeff Madsen, 2Sep2009.) (Proof shortened by
Mario Carneiro, 8Sep2015.)



Theorem  iccf1o 8872* 
Describe a bijection from to an arbitrary nontrivial
closed interval . (Contributed by Mario Carneiro,
8Sep2015.)



Theorem  unitssre 8873 
is a subset of the reals.
(Contributed by David Moews,
28Feb2017.)



3.5.4 Finite intervals of integers


Syntax  cfz 8874 
Extend class notation to include the notation for a contiguous finite set
of integers. Read " " as "the set of
integers from to
inclusive."



Definition  dffz 8875* 
Define an operation that produces a finite set of sequential integers.
Read " " as "the set of integers from
to
inclusive." See fzval 8876 for its value and additional comments.
(Contributed by NM, 6Sep2005.)



Theorem  fzval 8876* 
The value of a finite set of sequential integers. E.g.,
means the set . A special case of this definition
(starting at 1) appears as Definition 112.1 of [Gleason] p. 141, where
_k means our
; he calls these sets segments of the
integers. (Contributed by NM, 6Sep2005.) (Revised by Mario Carneiro,
3Nov2013.)



Theorem  fzval2 8877 
An alternative way of expressing a finite set of sequential integers.
(Contributed by Mario Carneiro, 3Nov2013.)



Theorem  fzf 8878 
Establish the domain and codomain of the finite integer sequence
function. (Contributed by Scott Fenton, 8Aug2013.) (Revised by Mario
Carneiro, 16Nov2013.)



Theorem  elfz1 8879 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.)



Theorem  elfz 8880 
Membership in a finite set of sequential integers. (Contributed by NM,
29Sep2005.)



Theorem  elfz2 8881 
Membership in a finite set of sequential integers. We use the fact that
an operation's value is empty outside of its domain to show
and . (Contributed by NM, 6Sep2005.)
(Revised by Mario
Carneiro, 28Apr2015.)



Theorem  elfz5 8882 
Membership in a finite set of sequential integers. (Contributed by NM,
26Dec2005.)



Theorem  elfz4 8883 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.) (Revised by Mario Carneiro, 28Apr2015.)



Theorem  elfzuzb 8884 
Membership in a finite set of sequential integers in terms of sets of
upper integers. (Contributed by NM, 18Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)



Theorem  eluzfz 8885 
Membership in a finite set of sequential integers. (Contributed by NM,
4Oct2005.) (Revised by Mario Carneiro, 28Apr2015.)



Theorem  elfzuz 8886 
A member of a finite set of sequential integers belongs to an upper set of
integers. (Contributed by NM, 17Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  elfzuz3 8887 
Membership in a finite set of sequential integers implies membership in an
upper set of integers. (Contributed by NM, 28Sep2005.) (Revised by
Mario Carneiro, 28Apr2015.)



Theorem  elfzel2 8888 
Membership in a finite set of sequential integer implies the upper bound
is an integer. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)



Theorem  elfzel1 8889 
Membership in a finite set of sequential integer implies the lower bound
is an integer. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)



Theorem  elfzelz 8890 
A member of a finite set of sequential integer is an integer.
(Contributed by NM, 6Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  elfzle1 8891 
A member of a finite set of sequential integer is greater than or equal to
the lower bound. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)



Theorem  elfzle2 8892 
A member of a finite set of sequential integer is less than or equal to
the upper bound. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)



Theorem  elfzuz2 8893 
Implication of membership in a finite set of sequential integers.
(Contributed by NM, 20Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  elfzle3 8894 
Membership in a finite set of sequential integer implies the bounds are
comparable. (Contributed by NM, 18Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)



Theorem  eluzfz1 8895 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  eluzfz2 8896 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 13Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  eluzfz2b 8897 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 14Sep2005.)



Theorem  elfz3 8898 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21Jul2005.)



Theorem  elfz1eq 8899 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19Sep2005.)



Theorem  elfzubelfz 8900 
If there is a member in a finite set of sequential integers, the upper
bound is also a member of this finite set of sequential integers.
(Contributed by Alexander van der Vekens, 31May2018.)

