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	elfzel1 10101	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 10102	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 10103	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 10104	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 10105	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 10106	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 10107	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 10108	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 10109	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 10110	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 10111	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 10112	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 10113	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 10114	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 10115*	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 10116	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 10117	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 10118	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 10119	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 10120	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 10121	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 10122	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 10123	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 10124	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 10125	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 10126	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 10127	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 10128	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 10129	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 10130	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 10131	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 10132	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 10133	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 10134	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 10135	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 ) )
Theorem	fzaddel 10136	Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( J e. ( M ... N ) <-> ( J + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
Theorem	fzsubel 10137	Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( J e. ( M ... N ) <-> ( J - K ) e. ( ( M - K ) ... ( N - K ) ) ) )
Theorem	fzopth 10138	A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) <-> ( M = J /\ N = K ) ) )
Theorem	fzass4 10139	Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( ( B e. ( A ... D ) /\ C e. ( B ... D ) ) <-> ( B e. ( A ... C ) /\ C e. ( A ... D ) ) )
Theorem	fzss1 10140	Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( ZZ>= ` M ) -> ( K ... N ) C_ ( M ... N ) )
Theorem	fzss2 10141	Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( N e. ( ZZ>= ` K ) -> ( M ... K ) C_ ( M ... N ) )
Theorem	fzssuz 10142	A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
|- ( M ... N ) C_ ( ZZ>= ` M )
Theorem	fzsn 10143	A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( M e. ZZ -> ( M ... M ) = { M } )
Theorem	fzssp1 10144	Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( M ... N ) C_ ( M ... ( N + 1 ) )
Theorem	fzssnn 10145	Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
|- ( M e. NN -> ( M ... N ) C_ NN )
Theorem	fzsuc 10146	Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
Theorem	fzpred 10147	Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
|- ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( { M } u. ( ( M + 1 ) ... N ) ) )
Theorem	fzpreddisj 10148	A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
|- ( N e. ( ZZ>= ` M ) -> ( { M } i^i ( ( M + 1 ) ... N ) ) = (/) )
Theorem	elfzp1 10149	Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... ( N + 1 ) ) <-> ( K e. ( M ... N ) \/ K = ( N + 1 ) ) ) )
Theorem	fzp1ss 10150	Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( M e. ZZ -> ( ( M + 1 ) ... N ) C_ ( M ... N ) )
Theorem	fzelp1 10151	Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> K e. ( M ... ( N + 1 ) ) )
Theorem	fzp1elp1 10152	Add one to an element of a finite set of integers. (Contributed by Jeff Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( M ... N ) -> ( K + 1 ) e. ( M ... ( N + 1 ) ) )
Theorem	fznatpl1 10153	Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
|- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ( I + 1 ) e. ( 1 ... N ) )
Theorem	fzpr 10154	A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( M e. ZZ -> ( M ... ( M + 1 ) ) = { M , ( M + 1 ) } )
Theorem	fztp 10155	A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
|- ( M e. ZZ -> ( M ... ( M + 2 ) ) = { M , ( M + 1 ) , ( M + 2 ) } )
Theorem	fzsuc2 10156	Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M - 1 ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
Theorem	fzp1disj 10157	( M ... ( N + 1 ) ) is the disjoint union of ( M ... N ) with { ( N + 1 ) }. (Contributed by Mario Carneiro, 7-Mar-2014.)
|- ( ( M ... N ) i^i { ( N + 1 ) } ) = (/)
Theorem	fzdifsuc 10158	Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
|- ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
Theorem	fzprval 10159*	Two ways of defining the first two values of a sequence on NN. (Contributed by NM, 5-Sep-2011.)
|- ( A. x e. ( 1 ... 2 ) ( F ` x ) = if ( x = 1 , A , B ) <-> ( ( F ` 1 ) = A /\ ( F ` 2 ) = B ) )
Theorem	fztpval 10160*	Two ways of defining the first three values of a sequence on NN. (Contributed by NM, 13-Sep-2011.)
|- ( A. x e. ( 1 ... 3 ) ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( ( F ` 1 ) = A /\ ( F ` 2 ) = B /\ ( F ` 3 ) = C ) )
Theorem	fzrev 10161	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( K e. ( ( J - N ) ... ( J - M ) ) <-> ( J - K ) e. ( M ... N ) ) )
Theorem	fzrev2 10162	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( K e. ( M ... N ) <-> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) ) )
Theorem	fzrev2i 10163	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
|- ( ( J e. ZZ /\ K e. ( M ... N ) ) -> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) )
Theorem	fzrev3 10164	The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
|- ( K e. ZZ -> ( K e. ( M ... N ) <-> ( ( M + N ) - K ) e. ( M ... N ) ) )
Theorem	fzrev3i 10165	The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
|- ( K e. ( M ... N ) -> ( ( M + N ) - K ) e. ( M ... N ) )
Theorem	fznn 10166	Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( K e. ( 1 ... N ) <-> ( K e. NN /\ K <_ N ) ) )
Theorem	elfz1b 10167	Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
|- ( N e. ( 1 ... M ) <-> ( N e. NN /\ M e. NN /\ N <_ M ) )
Theorem	elfzm11 10168	Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... ( N - 1 ) ) <-> ( K e. ZZ /\ M <_ K /\ K < N ) ) )
Theorem	uzsplit 10169	Express an upper integer set as the disjoint (see uzdisj 10170) union of the first N values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
|- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` M ) = ( ( M ... ( N - 1 ) ) u. ( ZZ>= ` N ) ) )
Theorem	uzdisj 10170	The first N elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)
|- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) = (/)
Theorem	fseq1p1m1 10171	Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
|- H = { <. ( N + 1 ) , B >. }   =>    |- ( N e. NN0 -> ( ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) <-> ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) )
Theorem	fseq1m1p1 10172	Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
|- H = { <. N , B >. }   =>    |- ( N e. NN -> ( ( F : ( 1 ... ( N - 1 ) ) --> A /\ B e. A /\ G = ( F u. H ) ) <-> ( G : ( 1 ... N ) --> A /\ ( G ` N ) = B /\ F = ( G |` ( 1 ... ( N - 1 ) ) ) ) ) )
Theorem	fz1sbc 10173*	Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
|- ( N e. ZZ -> ( A. k e. ( N ... N ) ph <-> [. N / k ]. ph ) )
Theorem	elfzp1b 10174	An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 0 ... ( N - 1 ) ) <-> ( K + 1 ) e. ( 1 ... N ) ) )
Theorem	elfzm1b 10175	An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) ) )
Theorem	elfzp12 10176	Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
|- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <-> ( K = M \/ K e. ( ( M + 1 ) ... N ) ) ) )
Theorem	fzm1 10177	Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
Theorem	fzneuz 10178	No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)
|- ( ( N e. ( ZZ>= ` M ) /\ K e. ZZ ) -> -. ( M ... N ) = ( ZZ>= ` K ) )
Theorem	fznuz 10179	Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)
|- ( K e. ( M ... N ) -> -. K e. ( ZZ>= ` ( N + 1 ) ) )
Theorem	uznfz 10180	Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
|- ( K e. ( ZZ>= ` N ) -> -. K e. ( M ... ( N - 1 ) ) )
Theorem	fzp1nel 10181	One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)
|- -. ( N + 1 ) e. ( M ... N )
Theorem	fzrevral 10182*	Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( A. j e. ( M ... N ) ph <-> A. k e. ( ( K - N ) ... ( K - M ) ) [. ( K - k ) / j ]. ph ) )
Theorem	fzrevral2 10183*	Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( A. j e. ( ( K - N ) ... ( K - M ) ) ph <-> A. k e. ( M ... N ) [. ( K - k ) / j ]. ph ) )
Theorem	fzrevral3 10184*	Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( A. j e. ( M ... N ) ph <-> A. k e. ( M ... N ) [. ( ( M + N ) - k ) / j ]. ph ) )
Theorem	fzshftral 10185*	Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( A. j e. ( M ... N ) ph <-> A. k e. ( ( M + K ) ... ( N + K ) ) [. ( k - K ) / j ]. ph ) )
Theorem	ige2m1fz1 10186	Membership of an integer greater than 1 decreased by 1 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. ( 1 ... N ) )
Theorem	ige2m1fz 10187	Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.)
|- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. ( 0 ... N ) )
Theorem	fz01or 10188	An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)
|- ( A e. ( 0 ... 1 ) <-> ( A = 0 \/ A = 1 ) )
4.5.5  Finite intervals of nonnegative integers Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound: ( 0 ... N ), usually abbreviated by "fz0".
Theorem	elfz2nn0 10189	Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( 0 ... N ) <-> ( K e. NN0 /\ N e. NN0 /\ K <_ N ) )
Theorem	fznn0 10190	Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
|- ( N e. NN0 -> ( K e. ( 0 ... N ) <-> ( K e. NN0 /\ K <_ N ) ) )
Theorem	elfznn0 10191	A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( 0 ... N ) -> K e. NN0 )
Theorem	elfz3nn0 10192	The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( 0 ... N ) -> N e. NN0 )
Theorem	fz0ssnn0 10193	Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
|- ( 0 ... N ) C_ NN0
Theorem	fz1ssfz0 10194	Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( 1 ... N ) C_ ( 0 ... N )
Theorem	0elfz 10195	0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.)
|- ( N e. NN0 -> 0 e. ( 0 ... N ) )
Theorem	nn0fz0 10196	A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.)
|- ( N e. NN0 <-> N e. ( 0 ... N ) )
Theorem	elfz0add 10197	An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( N e. ( 0 ... A ) -> N e. ( 0 ... ( A + B ) ) ) )
Theorem	fz0sn 10198	An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
|- ( 0 ... 0 ) = { 0 }
Theorem	fz0tp 10199	An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
|- ( 0 ... 2 ) = { 0 , 1 , 2 }
Theorem	fz0to3un2pr 10200	An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021.)
|- ( 0 ... 3 ) = ( { 0 , 1 } u. { 2 , 3 } )