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	iccmax 10301	The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
|- ( -oo [,] +oo ) = RR*
Theorem	ioopos 10302	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 10303	The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( 0 (,) +oo ) = RR+
Theorem	iooshf 10304	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 10305	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 10306	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 10307	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 10308	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 10309	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 10310	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 10311	An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
|- ( A (,) B ) C_ ( A [,] B )
Theorem	icossicc 10312	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 10313	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 10314	An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
|- ( A (,) B ) C_ ( A [,) B )
Theorem	iocssioo 10315	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 10316	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 10317	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 10318*	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 10319	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 10320	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 10321	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 10322	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 10323	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 10324	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 10325	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 10326*	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 10327	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 10328	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 10329	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 10330	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 10331	0 is a member of ( 0 [,) +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 e. ( 0 [,) +oo )
Theorem	0e0iccpnf 10332	0 is a member of ( 0 [,] +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 e. ( 0 [,] +oo )
Theorem	ge0addcl 10333	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 10334	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 10335	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 10336	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 10337	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 10338	Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- 0 e. ( 0 [,] 1 )
Theorem	1elunit 10339	One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- 1 e. ( 0 [,] 1 )
Theorem	iooneg 10340	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 10341	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 10342	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 10343*	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 10344	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 10345	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 10346	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 10347	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 10348	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 10349	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 10350	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 ) ) )
Theorem	iccdili 10351	Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. RR   &    |- B e. RR   &    |- R e. RR+   &    |- ( A x. R ) = C   &    |- ( B x. R ) = D   =>    |- ( X e. ( A [,] B ) -> ( X x. R ) e. ( C [,] D ) )
Theorem	icccntr 10352	Membership in a contracted 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	icccntri 10353	Membership in a contracted 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	divelunit 10354	A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) )
Theorem	lincmb01cmp 10355	A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)
|- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) )
Theorem	lincmble 10356	A linear combination of two reals which lies in the interval between them. Like lincmb01cmp 10355 but generalized to require merely A <_ B not A < B. (Contributed by Jim Kingdon, 13-May-2026.)
|- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) )
Theorem	iccf1o 10357*	Describe a bijection from [ 0 , 1 ] to an arbitrary nontrivial closed interval [ A , B ]. (Contributed by Mario Carneiro, 8-Sep-2015.)
|- F = ( x e. ( 0 [,] 1 ) |-> ( ( x x. B ) + ( ( 1 - x ) x. A ) ) )   =>    |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) /\ `' F = ( y e. ( A [,] B ) |-> ( ( y - A ) / ( B - A ) ) ) ) )
Theorem	unitssre 10358	( 0 [,] 1 ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
|- ( 0 [,] 1 ) C_ RR
Theorem	iccen 10359	Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
|- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( 0 [,] 1 ) ~~ ( A [,] B ) )
Theorem	zltaddlt1le 10360	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, 1-Jul-2021.)
|- ( ( M e. ZZ /\ N e. ZZ /\ A e. ( 0 (,) 1 ) ) -> ( ( M + A ) < N <-> ( M + A ) <_ N ) )
4.5.4  Finite intervals of integers
Syntax	cfz 10361	Extend class notation to include the notation for a contiguous finite set of integers. Read " M ... N " as "the set of integers from M to N inclusive". This symbol is also used informally in some comments to denote an ellipsis, e.g., A + A ^ 2 + ... + A ^ ( N - 1 ).
class ...
Definition	df-fz 10362*	Define an operation that produces a finite set of sequential integers. Read " M ... N " as "the set of integers from M to N inclusive". See fzval 10363 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
|- ... = ( m e. ZZ , n e. ZZ |-> { k e. ZZ | ( m <_ k /\ k <_ n ) } )
Theorem	fzval 10363*	The value of a finite set of sequential integers. E.g., 2 ... 5 means the set { 2 , 3 , 4 , 5 }. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where NNk means our 1 ... k; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = { k e. ZZ | ( M <_ k /\ k <_ N ) } )
Theorem	fzval2 10364	An alternate way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) = ( ( M [,] N ) i^i ZZ ) )
Theorem	fzf 10365	Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- ... : ( ZZ X. ZZ ) --> ~P ZZ
Theorem	elfz1 10366	Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> ( K e. ZZ /\ M <_ K /\ K <_ N ) ) )
Theorem	elfz 10367	Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
Theorem	elfz2 10368	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 M e. ZZ and N e. ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) <-> ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) ) )
Theorem	elfzd 10369	Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> M <_ K )   &    |- ( ph -> K <_ N )   =>    |- ( ph -> K e. ( M ... N ) )
Theorem	elfz5 10370	Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
|- ( ( K e. ( ZZ>= ` M ) /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> K <_ N ) )
Theorem	elfz4 10371	Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M <_ K /\ K <_ N ) ) -> K e. ( M ... N ) )
Theorem	elfzuzb 10372	Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) <-> ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) )
Theorem	eluzfz 10373	Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( K e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) -> K e. ( M ... N ) )
Theorem	elfzuz 10374	A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
Theorem	elfzuz3 10375	Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
Theorem	elfzel2 10376	Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> N e. ZZ )
Theorem	elfzel1 10377	Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> M e. ZZ )
Theorem	elfzelz 10378	A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> K e. ZZ )
Theorem	elfzelzd 10379	A member of a finite set of sequential integers is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> K e. ( M ... N ) )   =>    |- ( ph -> K e. ZZ )
Theorem	fzssz 10380	A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( M ... N ) C_ ZZ
Theorem	elfzle1 10381	A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> M <_ K )
Theorem	elfzle2 10382	A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> K <_ N )
Theorem	elfzuz2 10383	Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` M ) )
Theorem	elfzle3 10384	Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> M <_ N )
Theorem	eluzfz1 10385	Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
Theorem	eluzfz2 10386	Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
Theorem	eluzfz2b 10387	Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) <-> N e. ( M ... N ) )
Theorem	elfz3 10388	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
|- ( N e. ZZ -> N e. ( N ... N ) )
Theorem	elfz1eq 10389	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
|- ( K e. ( N ... N ) -> K = N )
Theorem	elfzubelfz 10390	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, 31-May-2018.)
|- ( K e. ( M ... N ) -> N e. ( M ... N ) )
Theorem	peano2fzr 10391	A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
|- ( ( K e. ( ZZ>= ` M ) /\ ( K + 1 ) e. ( M ... N ) ) -> K e. ( M ... N ) )
Theorem	fzm 10392*	Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
|- ( E. x x e. ( M ... N ) <-> N e. ( ZZ>= ` M ) )
Theorem	fztri3or 10393	Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
Theorem	fzdcel 10394	Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID K e. ( M ... N ) )
Theorem	fznlem 10395	A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M -> ( M ... N ) = (/) ) )
Theorem	fzn 10396	A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) )
Theorem	fzen 10397	A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M ... N ) ~~ ( ( M + K ) ... ( N + K ) ) )
Theorem	fz1n 10398	A 1-based finite set of sequential integers is empty iff it ends at index 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. NN0 -> ( ( 1 ... N ) = (/) <-> N = 0 ) )
Theorem	0fz1 10399	Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
|- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> ( F = (/) <-> N = 0 ) )
Theorem	fz10 10400	There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( 1 ... 0 ) = (/)