P. 102 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ioossicc 10101	An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
|- ( A (,) B ) C_ ( A [,] B )
Theorem	icossicc 10102	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 10103	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 10104	An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
|- ( A (,) B ) C_ ( A [,) B )
Theorem	iocssioo 10105	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 10106	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 10107	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 10108*	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 10109	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 10110	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 10111	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 10112	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 10113	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 10114	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 10115	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 10116*	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 10117	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 10118	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 10119	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 10120	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 10121	0 is a member of ( 0 [,) +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 e. ( 0 [,) +oo )
Theorem	0e0iccpnf 10122	0 is a member of ( 0 [,] +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 e. ( 0 [,] +oo )
Theorem	ge0addcl 10123	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 10124	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 10125	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 10126	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 10127	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 10128	Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- 0 e. ( 0 [,] 1 )
Theorem	1elunit 10129	One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
|- 1 e. ( 0 [,] 1 )
Theorem	iooneg 10130	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 10131	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 10132	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 10133*	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 10134	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 10135	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 10136	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 10137	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 10138	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 10139	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 10140	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 10141	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 10142	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 10143	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 10144	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 10145	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	iccf1o 10146*	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 10147	( 0 [,] 1 ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
|- ( 0 [,] 1 ) C_ RR
Theorem	iccen 10148	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 10149	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 10150	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 10151*	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 10152 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 10152*	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 10153	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 10154	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 10155	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 10156	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 10157	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 10158	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 10159	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 10160	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 10161	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 10162	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 10163	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 10164	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 10165	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 10166	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 10167	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 10168	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	elfzle1 10169	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 10170	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 10171	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 10172	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 10173	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 10174	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 10175	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 10176	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 10177	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 10178	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 10179	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 10180*	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 10181	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 10182	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 10183	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 10184	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 10185	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 10186	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 10187	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 10188	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 ) = (/)
Theorem	uzsubsubfz 10189	Membership of an integer greater than L decreased by ( L - M ) in an M based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
|- ( ( L e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` L ) ) -> ( N - ( L - M ) ) e. ( M ... N ) )
Theorem	uzsubsubfz1 10190	Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
|- ( ( L e. NN /\ N e. ( ZZ>= ` L ) ) -> ( N - ( L - 1 ) ) e. ( 1 ... N ) )
Theorem	ige3m2fz 10191	Membership of an integer greater than 2 decreased by 2 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
|- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. ( 1 ... N ) )
Theorem	fzsplit2 10192	Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
|- ( ( ( K + 1 ) e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` K ) ) -> ( M ... N ) = ( ( M ... K ) u. ( ( K + 1 ) ... N ) ) )
Theorem	fzsplit 10193	Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
|- ( K e. ( M ... N ) -> ( M ... N ) = ( ( M ... K ) u. ( ( K + 1 ) ... N ) ) )
Theorem	fzdisj 10194	Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( K < M -> ( ( J ... K ) i^i ( M ... N ) ) = (/) )
Theorem	fz01en 10195	0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ( N e. ZZ -> ( 0 ... ( N - 1 ) ) ~~ ( 1 ... N ) )
Theorem	elfznn 10196	A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)
|- ( K e. ( 1 ... N ) -> K e. NN )
Theorem	elfz1end 10197	A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( A e. NN <-> A e. ( 1 ... A ) )
Theorem	fz1ssnn 10198	A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( 1 ... A ) C_ NN
Theorem	fznn0sub 10199	Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> ( N - K ) e. NN0 )
Theorem	fzmmmeqm 10200	Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
|- ( M e. ( L ... N ) -> ( ( N - L ) - ( M - L ) ) = ( N - M ) )