P. 106 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzrev 10501	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 10502	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 10503	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 10504	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 10505	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 10506	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 10507	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 10508	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 10509	Express an upper integer set as the disjoint (see uzdisj 10510) 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 10510	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 10511	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 10512	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 10513*	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 10514	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 10515	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 10516	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 10517	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 10518	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 10519	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 10520	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 10521	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 10522*	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 10523*	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 10524*	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 10525*	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 10526	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 10527	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 10528	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 10529	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 10530	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 10531	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 10532	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 10533	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 10534	Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( 1 ... N ) C_ ( 0 ... N )
Theorem	0elfz 10535	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 10536	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 10537	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 10538	An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
|- ( 0 ... 0 ) = { 0 }
Theorem	fz0tp 10539	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 10540	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 } )
Theorem	fz0to4untppr 10541	An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
|- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
Theorem	elfz0ubfz0 10542	An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
|- ( ( K e. ( 0 ... N ) /\ L e. ( K ... N ) ) -> K e. ( 0 ... L ) )
Theorem	elfz0fzfz0 10543	A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.)
|- ( ( M e. ( 0 ... L ) /\ N e. ( L ... X ) ) -> M e. ( 0 ... N ) )
Theorem	fz0fzelfz0 10544	If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
|- ( ( N e. ( 0 ... R ) /\ M e. ( N ... R ) ) -> M e. ( 0 ... R ) )
Theorem	fznn0sub2 10545	Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) )
Theorem	uzsubfz0 10546	Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
|- ( ( L e. NN0 /\ N e. ( ZZ>= ` L ) ) -> ( N - L ) e. ( 0 ... N ) )
Theorem	fz0fzdiffz0 10547	The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
|- ( ( M e. ( 0 ... N ) /\ K e. ( M ... N ) ) -> ( K - M ) e. ( 0 ... N ) )
Theorem	elfzmlbm 10548	Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
|- ( K e. ( M ... N ) -> ( K - M ) e. ( 0 ... ( N - M ) ) )
Theorem	elfzmlbp 10549	Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
|- ( ( N e. ZZ /\ K e. ( M ... ( M + N ) ) ) -> ( K - M ) e. ( 0 ... N ) )
Theorem	fzctr 10550	Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
|- ( N e. NN0 -> N e. ( 0 ... ( 2 x. N ) ) )
Theorem	difelfzle 10551	The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
|- ( ( K e. ( 0 ... N ) /\ M e. ( 0 ... N ) /\ K <_ M ) -> ( M - K ) e. ( 0 ... N ) )
Theorem	difelfznle 10552	The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
|- ( ( K e. ( 0 ... N ) /\ M e. ( 0 ... N ) /\ -. K <_ M ) -> ( ( M + N ) - K ) e. ( 0 ... N ) )
Theorem	nn0split 10553	Express the set of nonnegative integers as the disjoint (see nn0disj 10555) union of the first N + 1 values and the rest. (Contributed by AV, 8-Nov-2019.)
|- ( N e. NN0 -> NN0 = ( ( 0 ... N ) u. ( ZZ>= ` ( N + 1 ) ) ) )
Theorem	nnsplit 10554	Express the set of positive integers as the disjoint union of the first N values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- ( N e. NN -> NN = ( ( 1 ... N ) u. ( ZZ>= ` ( N + 1 ) ) ) )
Theorem	nn0disj 10555	The first N + 1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
|- ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) = (/)
Theorem	1fv 10556	A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
|- ( ( N e. V /\ P = { <. 0 , N >. } ) -> ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) )
Theorem	4fvwrd4 10557*	The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
|- ( ( L e. ( ZZ>= ` 3 ) /\ P : ( 0 ... L ) --> V ) -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( P ` 0 ) = a /\ ( P ` 1 ) = b ) /\ ( ( P ` 2 ) = c /\ ( P ` 3 ) = d ) ) )
Theorem	2ffzeq 10558*	Two functions over 0 based finite set of sequential integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
|- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F = P <-> ( M = N /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) )
4.5.6  Half-open integer ranges
Syntax	cfzo 10559	Syntax for half-open integer ranges.
class ..^
Definition	df-fzo 10560*	Define a function generating sets of integers using a half-open range. Read ( M..^ N ) as the integers from M up to, but not including, N; contrast with ( M ... N ) df-fz 10422, which includes N. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10596 with fzsplit 10466, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ..^ = ( m e. ZZ , n e. ZZ |-> ( m ... ( n - 1 ) ) )
Theorem	fzof 10561	Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ..^ : ( ZZ X. ZZ ) --> ~P ZZ
Theorem	elfzoel1 10562	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> B e. ZZ )
Theorem	elfzoel2 10563	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> C e. ZZ )
Theorem	elfzoelz 10564	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( A e. ( B..^ C ) -> A e. ZZ )
Theorem	fzoval 10565	Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( N e. ZZ -> ( M..^ N ) = ( M ... ( N - 1 ) ) )
Theorem	elfzo 10566	Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M..^ N ) <-> ( M <_ K /\ K < N ) ) )
Theorem	elfzo2 10567	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) <-> ( K e. ( ZZ>= ` M ) /\ N e. ZZ /\ K < N ) )
Theorem	elfzouz 10568	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) -> K e. ( ZZ>= ` M ) )
Theorem	nelfzo 10569	An integer not being a member of a half-open finite set of integers. (Contributed by AV, 29-Apr-2020.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e/ ( M..^ N ) <-> ( K < M \/ N <_ K ) ) )
Theorem	fzodcel 10570	Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID K e. ( M..^ N ) )
Theorem	fzolb 10571	The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with M < N. This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate M e. ( ZZ>= ` N ). (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( M e. ( M..^ N ) <-> ( M e. ZZ /\ N e. ZZ /\ M < N ) )
Theorem	fzolb2 10572	The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with M < N. This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate M e. ( ZZ>= ` N ). (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. ( M..^ N ) <-> M < N ) )
Theorem	elfzole1 10573	A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( K e. ( M..^ N ) -> M <_ K )
Theorem	elfzolt2 10574	A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( K e. ( M..^ N ) -> K < N )
Theorem	elfzolt3 10575	Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( K e. ( M..^ N ) -> M < N )
Theorem	elfzolt2b 10576	A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) -> K e. ( K..^ N ) )
Theorem	elfzolt3b 10577	Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) -> M e. ( M..^ N ) )
Theorem	fzonel 10578	A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
|- -. B e. ( A..^ B )
Theorem	elfzouz2 10579	The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) -> N e. ( ZZ>= ` K ) )
Theorem	elfzofz 10580	A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( K e. ( M..^ N ) -> K e. ( M ... N ) )
Theorem	elfzo3 10581	Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp. K e. ( ZZ>= ` M ) <-> M <_ K, K e. ( K..^ N ) <-> K < N. (Contributed by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( M..^ N ) <-> ( K e. ( ZZ>= ` M ) /\ K e. ( K..^ N ) ) )
Theorem	fzom 10582*	A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
|- ( E. x x e. ( M..^ N ) <-> M e. ( M..^ N ) )
Theorem	fzossfz 10583	A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|- ( A..^ B ) C_ ( A ... B )
Theorem	fzon 10584	A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N <_ M <-> ( M..^ N ) = (/) ) )
Theorem	fzo0n 10585	A half-open range of nonnegative integers is empty iff the upper bound is not positive. (Contributed by AV, 2-May-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N <_ M <-> ( 0..^ ( N - M ) ) = (/) ) )
Theorem	fzonlt0 10586	A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. M < N <-> ( M..^ N ) = (/) ) )
Theorem	fzo0 10587	Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|- ( A..^ A ) = (/)
Theorem	fzonnsub 10588	If K < N then N - K is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
|- ( K e. ( M..^ N ) -> ( N - K ) e. NN )
Theorem	fzonnsub2 10589	If M < N then N - M is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
|- ( K e. ( M..^ N ) -> ( N - M ) e. NN )
Theorem	fzoss1 10590	Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|- ( K e. ( ZZ>= ` M ) -> ( K..^ N ) C_ ( M..^ N ) )
Theorem	fzoss2 10591	Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|- ( N e. ( ZZ>= ` K ) -> ( M..^ K ) C_ ( M..^ N ) )
Theorem	fzossrbm1 10592	Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
|- ( N e. ZZ -> ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) )
Theorem	fzo0ss1 10593	Subset relationship for half-open integer ranges with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
|- ( 1..^ N ) C_ ( 0..^ N )
Theorem	fzossnn0 10594	A half-open integer range starting at a nonnegative integer is a subset of the nonnegative integers. (Contributed by Alexander van der Vekens, 13-May-2018.)
|- ( M e. NN0 -> ( M..^ N ) C_ NN0 )
Theorem	fzospliti 10595	One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( ( A e. ( B..^ C ) /\ D e. ZZ ) -> ( A e. ( B..^ D ) \/ A e. ( D..^ C ) ) )
Theorem	fzosplit 10596	Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( D e. ( B ... C ) -> ( B..^ C ) = ( ( B..^ D ) u. ( D..^ C ) ) )
Theorem	fzodisj 10597	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
|- ( ( A..^ B ) i^i ( B..^ C ) ) = (/)
Theorem	fzouzsplit 10598	Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
|- ( B e. ( ZZ>= ` A ) -> ( ZZ>= ` A ) = ( ( A..^ B ) u. ( ZZ>= ` B ) ) )
Theorem	fzouzdisj 10599	A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.)
|- ( ( A..^ B ) i^i ( ZZ>= ` B ) ) = (/)
Theorem	fzoun 10600	A half-open integer range as union of two half-open integer ranges. (Contributed by AV, 23-Apr-2022.)
|- ( ( B e. ( ZZ>= ` A ) /\ C e. NN0 ) -> ( A..^ ( B + C ) ) = ( ( A..^ B ) u. ( B..^ ( B + C ) ) ) )