Type  Label  Description 
Statement 

Theorem  iccleub 8901 
An element of a closed interval is less than or equal to its upper bound.
(Contributed by Jeff Hankins, 14Jul2009.)



Theorem  iccgelb 8902 
An element of a closed interval is more than or equal to its lower bound
(Contributed by Thierry Arnoux, 23Dec2016.)



Theorem  elioo5 8903 
Membership in an open interval of extended reals. (Contributed by NM,
17Aug2008.)



Theorem  elioo4g 8904 
Membership in an open interval of extended reals. (Contributed by NM,
8Jun2007.) (Revised by Mario Carneiro, 28Apr2015.)



Theorem  ioossre 8905 
An open interval is a set of reals. (Contributed by NM,
31May2007.)



Theorem  elioc2 8906 
Membership in an openbelow, closedabove real interval. (Contributed by
Paul Chapman, 30Dec2007.) (Revised by Mario Carneiro, 14Jun2014.)



Theorem  elico2 8907 
Membership in a closedbelow, openabove real interval. (Contributed by
Paul Chapman, 21Jan2008.) (Revised by Mario Carneiro, 14Jun2014.)



Theorem  elicc2 8908 
Membership in a closed real interval. (Contributed by Paul Chapman,
21Sep2007.) (Revised by Mario Carneiro, 14Jun2014.)



Theorem  elicc2i 8909 
Inference for membership in a closed interval. (Contributed by Scott
Fenton, 3Jun2013.)



Theorem  elicc4 8910 
Membership in a closed real interval. (Contributed by Stefan O'Rear,
16Nov2014.) (Proof shortened by Mario Carneiro, 1Jan2017.)



Theorem  iccss 8911 
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 8912 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20Feb2015.)



Theorem  icossico 8913 
Condition for a closedbelow, openabove interval to be a subset of a
closedbelow, openabove interval. (Contributed by Thierry Arnoux,
21Sep2017.)



Theorem  iccss2 8914 
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 8915 
Condition for a closed interval to be a subset of a halfopen interval.
(Contributed by Mario Carneiro, 9Sep2015.)



Theorem  iccssioo2 8916 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20Feb2015.)



Theorem  iccssico2 8917 
Condition for a closed interval to be a subset of a closedbelow,
openabove interval. (Contributed by Mario Carneiro, 20Feb2015.)



Theorem  ioomax 8918 
The open interval from minus to plus infinity. (Contributed by NM,
6Feb2007.)



Theorem  iccmax 8919 
The closed interval from minus to plus infinity. (Contributed by Mario
Carneiro, 4Jul2014.)



Theorem  ioopos 8920 
The set of positive reals expressed as an open interval. (Contributed by
NM, 7May2007.)



Theorem  ioorp 8921 
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25Nov2007.)



Theorem  iooshf 8922 
Shift the arguments of the open interval function. (Contributed by NM,
17Aug2008.)



Theorem  iocssre 8923 
A closedabove interval with real upper bound is a set of reals.
(Contributed by FL, 29May2014.)



Theorem  icossre 8924 
A closedbelow interval with real lower bound is a set of reals.
(Contributed by Mario Carneiro, 14Jun2014.)



Theorem  iccssre 8925 
A closed real interval is a set of reals. (Contributed by FL,
6Jun2007.) (Proof shortened by Paul Chapman, 21Jan2008.)



Theorem  iccssxr 8926 
A closed interval is a set of extended reals. (Contributed by FL,
28Jul2008.) (Revised by Mario Carneiro, 4Jul2014.)



Theorem  iocssxr 8927 
An openbelow, closedabove interval is a subset of the extended reals.
(Contributed by FL, 29May2014.) (Revised by Mario Carneiro,
4Jul2014.)



Theorem  icossxr 8928 
A closedbelow, openabove interval is a subset of the extended reals.
(Contributed by FL, 29May2014.) (Revised by Mario Carneiro,
4Jul2014.)



Theorem  ioossicc 8929 
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18Oct2007.)



Theorem  icossicc 8930 
A closedbelow, openabove interval is a subset of its closure.
(Contributed by Thierry Arnoux, 25Oct2016.)



Theorem  iocssicc 8931 
A closedabove, openbelow interval is a subset of its closure.
(Contributed by Thierry Arnoux, 1Apr2017.)



Theorem  ioossico 8932 
An open interval is a subset of its closurebelow. (Contributed by
Thierry Arnoux, 3Mar2017.)



Theorem  iocssioo 8933 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29Mar2017.)



Theorem  icossioo 8934 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29Mar2017.)



Theorem  ioossioo 8935 
Condition for an open interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 26Sep2017.)



Theorem  iccsupr 8936* 
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 8937 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)



Theorem  elioomnf 8938 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)



Theorem  elicopnf 8939 
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16Sep2014.)



Theorem  repos 8940 
Two ways of saying that a real number is positive. (Contributed by NM,
7May2007.)



Theorem  ioof 8941 
The set of open intervals of extended reals maps to subsets of reals.
(Contributed by NM, 7Feb2007.) (Revised by Mario Carneiro,
16Nov2013.)



Theorem  iccf 8942 
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 8943 
The union of the range of the open interval function. (Contributed by
NM, 7May2007.) (Revised by Mario Carneiro, 30Jan2014.)



Theorem  dfioo2 8944* 
Alternate definition of the set of open intervals of extended reals.
(Contributed by NM, 1Mar2007.) (Revised by Mario Carneiro,
1Sep2015.)



Theorem  ioorebasg 8945 
Open intervals are elements of the set of all open intervals.
(Contributed by Jim Kingdon, 4Apr2020.)



Theorem  elrege0 8946 
The predicate "is a nonnegative real". (Contributed by Jeff Madsen,
2Sep2009.) (Proof shortened by Mario Carneiro, 18Jun2014.)



Theorem  rge0ssre 8947 
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9Sep2018.) (Proof shortened by AV, 8Sep2019.)



Theorem  elxrge0 8948 
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28Jun2014.)



Theorem  0e0icopnf 8949 
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8Dec2018.)



Theorem  0e0iccpnf 8950 
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8Dec2018.)



Theorem  ge0addcl 8951 
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19Jun2014.)



Theorem  ge0mulcl 8952 
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19Jun2014.)



Theorem  lbicc2 8953 
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 8954 
The upper bound of a closed interval is a member of it. (Contributed by
Paul Chapman, 26Nov2007.) (Revised by FL, 29May2014.)



Theorem  0elunit 8955 
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)



Theorem  1elunit 8956 
One is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)



Theorem  iooneg 8957 
Membership in a negated open real interval. (Contributed by Paul Chapman,
26Nov2007.)



Theorem  iccneg 8958 
Membership in a negated closed real interval. (Contributed by Paul
Chapman, 26Nov2007.)



Theorem  icoshft 8959 
A shifted real is a member of a shifted, closedbelow, openabove real
interval. (Contributed by Paul Chapman, 25Mar2008.)



Theorem  icoshftf1o 8960* 
Shifting a closedbelow, openabove interval is onetoone onto.
(Contributed by Paul Chapman, 25Mar2008.) (Proof shortened by Mario
Carneiro, 1Sep2015.)



Theorem  icodisj 8961 
Endtoend closedbelow, openabove real intervals are disjoint.
(Contributed by Mario Carneiro, 16Jun2014.)



Theorem  ioodisj 8962 
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 8963 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftri 8964 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftl 8965 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftli 8966 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccdil 8967 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccdili 8968 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  icccntr 8969 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  icccntri 8970 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  divelunit 8971 
A condition for a ratio to be a member of the closed unit. (Contributed
by Scott Fenton, 11Jun2013.)



Theorem  lincmb01cmp 8972 
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 8973* 
Describe a bijection from to an arbitrary nontrivial
closed interval . (Contributed by Mario Carneiro,
8Sep2015.)



Theorem  unitssre 8974 
is a subset of the reals.
(Contributed by David Moews,
28Feb2017.)



Theorem  zltaddlt1le 8975 
The sum of an integer and a real number between 0 and 1 is less than or
equal to a second integer iff the sum is less than the second integer.
(Contributed by AV, 1Jul2021.)



3.5.4 Finite intervals of integers


Syntax  cfz 8976 
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 8977* 
Define an operation that produces a finite set of sequential integers.
Read " " as "the set of integers from
to
inclusive." See fzval 8978 for its value and additional comments.
(Contributed by NM, 6Sep2005.)



Theorem  fzval 8978* 
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 8979 
An alternative way of expressing a finite set of sequential integers.
(Contributed by Mario Carneiro, 3Nov2013.)



Theorem  fzf 8980 
Establish the domain and codomain of the finite integer sequence
function. (Contributed by Scott Fenton, 8Aug2013.) (Revised by Mario
Carneiro, 16Nov2013.)



Theorem  elfz1 8981 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.)



Theorem  elfz 8982 
Membership in a finite set of sequential integers. (Contributed by NM,
29Sep2005.)



Theorem  elfz2 8983 
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 8984 
Membership in a finite set of sequential integers. (Contributed by NM,
26Dec2005.)



Theorem  elfz4 8985 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.) (Revised by Mario Carneiro, 28Apr2015.)



Theorem  elfzuzb 8986 
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 8987 
Membership in a finite set of sequential integers. (Contributed by NM,
4Oct2005.) (Revised by Mario Carneiro, 28Apr2015.)



Theorem  elfzuz 8988 
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 8989 
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 8990 
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 8991 
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 8992 
A member of a finite set of sequential integer is an integer.
(Contributed by NM, 6Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  elfzle1 8993 
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 8994 
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 8995 
Implication of membership in a finite set of sequential integers.
(Contributed by NM, 20Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  elfzle3 8996 
Membership in a finite set of sequential integer implies the bounds are
comparable. (Contributed by NM, 18Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)



Theorem  eluzfz1 8997 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  eluzfz2 8998 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 13Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)



Theorem  eluzfz2b 8999 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 14Sep2005.)



Theorem  elfz3 9000 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21Jul2005.)

