Type  Label  Description 
Statement 

Theorem  ubioog 9301 
An open interval does not contain its right endpoint. (Contributed by
Jim Kingdon, 30Mar2020.)



Theorem  iooval2 9302* 
Value of the open interval function. (Contributed by NM, 6Feb2007.)
(Revised by Mario Carneiro, 3Nov2013.)



Theorem  iooss1 9303 
Subset relationship for open intervals of extended reals. (Contributed
by NM, 7Feb2007.) (Revised by Mario Carneiro, 20Feb2015.)



Theorem  iooss2 9304 
Subset relationship for open intervals of extended reals. (Contributed
by NM, 7Feb2007.) (Revised by Mario Carneiro, 3Nov2013.)



Theorem  iocval 9305* 
Value of the openbelow, closedabove interval function. (Contributed
by NM, 24Dec2006.) (Revised by Mario Carneiro, 3Nov2013.)



Theorem  icoval 9306* 
Value of the closedbelow, openabove interval function. (Contributed
by NM, 24Dec2006.) (Revised by Mario Carneiro, 3Nov2013.)



Theorem  iccval 9307* 
Value of the closed interval function. (Contributed by NM,
24Dec2006.) (Revised by Mario Carneiro, 3Nov2013.)



Theorem  elioo2 9308 
Membership in an open interval of extended reals. (Contributed by NM,
6Feb2007.)



Theorem  elioc1 9309 
Membership in an openbelow, closedabove interval of extended reals.
(Contributed by NM, 24Dec2006.) (Revised by Mario Carneiro,
3Nov2013.)



Theorem  elico1 9310 
Membership in a closedbelow, openabove interval of extended reals.
(Contributed by NM, 24Dec2006.) (Revised by Mario Carneiro,
3Nov2013.)



Theorem  elicc1 9311 
Membership in a closed interval of extended reals. (Contributed by NM,
24Dec2006.) (Revised by Mario Carneiro, 3Nov2013.)



Theorem  iccid 9312 
A closed interval with identical lower and upper bounds is a singleton.
(Contributed by Jeff Hankins, 13Jul2009.)



Theorem  icc0r 9313 
An empty closed interval of extended reals. (Contributed by Jim
Kingdon, 30Mar2020.)



Theorem  eliooxr 9314 
An inhabited open interval spans an interval of extended reals.
(Contributed by NM, 17Aug2008.)



Theorem  eliooord 9315 
Ordering implied by a member of an open interval of reals. (Contributed
by NM, 17Aug2008.) (Revised by Mario Carneiro, 9May2014.)



Theorem  ubioc1 9316 
The upper bound belongs to an openbelow, closedabove interval. See
ubicc2 9371. (Contributed by FL, 29May2014.)



Theorem  lbico1 9317 
The lower bound belongs to a closedbelow, openabove interval. See
lbicc2 9370. (Contributed by FL, 29May2014.)



Theorem  iccleub 9318 
An element of a closed interval is less than or equal to its upper bound.
(Contributed by Jeff Hankins, 14Jul2009.)



Theorem  iccgelb 9319 
An element of a closed interval is more than or equal to its lower bound
(Contributed by Thierry Arnoux, 23Dec2016.)



Theorem  elioo5 9320 
Membership in an open interval of extended reals. (Contributed by NM,
17Aug2008.)



Theorem  elioo4g 9321 
Membership in an open interval of extended reals. (Contributed by NM,
8Jun2007.) (Revised by Mario Carneiro, 28Apr2015.)



Theorem  ioossre 9322 
An open interval is a set of reals. (Contributed by NM,
31May2007.)



Theorem  elioc2 9323 
Membership in an openbelow, closedabove real interval. (Contributed by
Paul Chapman, 30Dec2007.) (Revised by Mario Carneiro, 14Jun2014.)



Theorem  elico2 9324 
Membership in a closedbelow, openabove real interval. (Contributed by
Paul Chapman, 21Jan2008.) (Revised by Mario Carneiro, 14Jun2014.)



Theorem  elicc2 9325 
Membership in a closed real interval. (Contributed by Paul Chapman,
21Sep2007.) (Revised by Mario Carneiro, 14Jun2014.)



Theorem  elicc2i 9326 
Inference for membership in a closed interval. (Contributed by Scott
Fenton, 3Jun2013.)



Theorem  elicc4 9327 
Membership in a closed real interval. (Contributed by Stefan O'Rear,
16Nov2014.) (Proof shortened by Mario Carneiro, 1Jan2017.)



Theorem  iccss 9328 
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 9329 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20Feb2015.)



Theorem  icossico 9330 
Condition for a closedbelow, openabove interval to be a subset of a
closedbelow, openabove interval. (Contributed by Thierry Arnoux,
21Sep2017.)



Theorem  iccss2 9331 
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 9332 
Condition for a closed interval to be a subset of a halfopen interval.
(Contributed by Mario Carneiro, 9Sep2015.)



Theorem  iccssioo2 9333 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20Feb2015.)



Theorem  iccssico2 9334 
Condition for a closed interval to be a subset of a closedbelow,
openabove interval. (Contributed by Mario Carneiro, 20Feb2015.)



Theorem  ioomax 9335 
The open interval from minus to plus infinity. (Contributed by NM,
6Feb2007.)



Theorem  iccmax 9336 
The closed interval from minus to plus infinity. (Contributed by Mario
Carneiro, 4Jul2014.)



Theorem  ioopos 9337 
The set of positive reals expressed as an open interval. (Contributed by
NM, 7May2007.)



Theorem  ioorp 9338 
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25Nov2007.)



Theorem  iooshf 9339 
Shift the arguments of the open interval function. (Contributed by NM,
17Aug2008.)



Theorem  iocssre 9340 
A closedabove interval with real upper bound is a set of reals.
(Contributed by FL, 29May2014.)



Theorem  icossre 9341 
A closedbelow interval with real lower bound is a set of reals.
(Contributed by Mario Carneiro, 14Jun2014.)



Theorem  iccssre 9342 
A closed real interval is a set of reals. (Contributed by FL,
6Jun2007.) (Proof shortened by Paul Chapman, 21Jan2008.)



Theorem  iccssxr 9343 
A closed interval is a set of extended reals. (Contributed by FL,
28Jul2008.) (Revised by Mario Carneiro, 4Jul2014.)



Theorem  iocssxr 9344 
An openbelow, closedabove interval is a subset of the extended reals.
(Contributed by FL, 29May2014.) (Revised by Mario Carneiro,
4Jul2014.)



Theorem  icossxr 9345 
A closedbelow, openabove interval is a subset of the extended reals.
(Contributed by FL, 29May2014.) (Revised by Mario Carneiro,
4Jul2014.)



Theorem  ioossicc 9346 
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18Oct2007.)



Theorem  icossicc 9347 
A closedbelow, openabove interval is a subset of its closure.
(Contributed by Thierry Arnoux, 25Oct2016.)



Theorem  iocssicc 9348 
A closedabove, openbelow interval is a subset of its closure.
(Contributed by Thierry Arnoux, 1Apr2017.)



Theorem  ioossico 9349 
An open interval is a subset of its closurebelow. (Contributed by
Thierry Arnoux, 3Mar2017.)



Theorem  iocssioo 9350 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29Mar2017.)



Theorem  icossioo 9351 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29Mar2017.)



Theorem  ioossioo 9352 
Condition for an open interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 26Sep2017.)



Theorem  iccsupr 9353* 
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 9354 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)



Theorem  elioomnf 9355 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)



Theorem  elicopnf 9356 
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16Sep2014.)



Theorem  repos 9357 
Two ways of saying that a real number is positive. (Contributed by NM,
7May2007.)



Theorem  ioof 9358 
The set of open intervals of extended reals maps to subsets of reals.
(Contributed by NM, 7Feb2007.) (Revised by Mario Carneiro,
16Nov2013.)



Theorem  iccf 9359 
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 9360 
The union of the range of the open interval function. (Contributed by
NM, 7May2007.) (Revised by Mario Carneiro, 30Jan2014.)



Theorem  dfioo2 9361* 
Alternate definition of the set of open intervals of extended reals.
(Contributed by NM, 1Mar2007.) (Revised by Mario Carneiro,
1Sep2015.)



Theorem  ioorebasg 9362 
Open intervals are elements of the set of all open intervals.
(Contributed by Jim Kingdon, 4Apr2020.)



Theorem  elrege0 9363 
The predicate "is a nonnegative real". (Contributed by Jeff Madsen,
2Sep2009.) (Proof shortened by Mario Carneiro, 18Jun2014.)



Theorem  rge0ssre 9364 
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9Sep2018.) (Proof shortened by AV, 8Sep2019.)



Theorem  elxrge0 9365 
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28Jun2014.)



Theorem  0e0icopnf 9366 
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8Dec2018.)



Theorem  0e0iccpnf 9367 
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8Dec2018.)



Theorem  ge0addcl 9368 
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19Jun2014.)



Theorem  ge0mulcl 9369 
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19Jun2014.)



Theorem  lbicc2 9370 
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 9371 
The upper bound of a closed interval is a member of it. (Contributed by
Paul Chapman, 26Nov2007.) (Revised by FL, 29May2014.)



Theorem  0elunit 9372 
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)



Theorem  1elunit 9373 
One is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)



Theorem  iooneg 9374 
Membership in a negated open real interval. (Contributed by Paul Chapman,
26Nov2007.)



Theorem  iccneg 9375 
Membership in a negated closed real interval. (Contributed by Paul
Chapman, 26Nov2007.)



Theorem  icoshft 9376 
A shifted real is a member of a shifted, closedbelow, openabove real
interval. (Contributed by Paul Chapman, 25Mar2008.)



Theorem  icoshftf1o 9377* 
Shifting a closedbelow, openabove interval is onetoone onto.
(Contributed by Paul Chapman, 25Mar2008.) (Proof shortened by Mario
Carneiro, 1Sep2015.)



Theorem  icodisj 9378 
Endtoend closedbelow, openabove real intervals are disjoint.
(Contributed by Mario Carneiro, 16Jun2014.)



Theorem  ioodisj 9379 
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 9380 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftri 9381 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftl 9382 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccshftli 9383 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccdil 9384 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  iccdili 9385 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  icccntr 9386 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  icccntri 9387 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  divelunit 9388 
A condition for a ratio to be a member of the closed unit. (Contributed
by Scott Fenton, 11Jun2013.)



Theorem  lincmb01cmp 9389 
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 9390* 
Describe a bijection from to an arbitrary nontrivial
closed interval . (Contributed by Mario Carneiro,
8Sep2015.)



Theorem  unitssre 9391 
is a subset of the reals.
(Contributed by David Moews,
28Feb2017.)



Theorem  zltaddlt1le 9392 
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 9393 
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 9394* 
Define an operation that produces a finite set of sequential integers.
Read " " as "the set of integers from
to
inclusive." See fzval 9395 for its value and additional comments.
(Contributed by NM, 6Sep2005.)



Theorem  fzval 9395* 
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 9396 
An alternate way of expressing a finite set of sequential integers.
(Contributed by Mario Carneiro, 3Nov2013.)



Theorem  fzf 9397 
Establish the domain and codomain of the finite integer sequence
function. (Contributed by Scott Fenton, 8Aug2013.) (Revised by Mario
Carneiro, 16Nov2013.)



Theorem  elfz1 9398 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.)



Theorem  elfz 9399 
Membership in a finite set of sequential integers. (Contributed by NM,
29Sep2005.)



Theorem  elfz2 9400 
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.)

