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Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfzopth 9901 A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzass4 9902 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Theoremfzss1 9903 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzss2 9904 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfzssuz 9905 A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)

Theoremfzsn 9906 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzssp1 9907 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzssnn 9908 Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)

Theoremfzsuc 9909 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.)

Theoremfzpred 9910 Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)

Theoremfzpreddisj 9911 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)

Theoremelfzp1 9912 Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzp1ss 9913 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzelp1 9914 Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzp1elp1 9915 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.)

Theoremfznatpl1 9916 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)

Theoremfzpr 9917 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfztp 9918 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)

Theoremfzsuc2 9919 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzp1disj 9920 is the disjoint union of with . (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzdifsuc 9921 Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)

Theoremfzprval 9922* Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.)

Theoremfztpval 9923* Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.)

Theoremfzrev 9924 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2 9925 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2i 9926 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev3 9927 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzrev3i 9928 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfznn 9929 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)

Theoremelfz1b 9930 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)

Theoremelfzm11 9931 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremuzsplit 9932 Express an upper integer set as the disjoint (see uzdisj 9933) union of the first values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremuzdisj 9933 The first elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremfseq1p1m1 9934 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.)

Theoremfseq1m1p1 9935 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremfz1sbc 9936* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)

Theoremelfzp1b 9937 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.)

Theoremelfzm1b 9938 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.)

Theoremelfzp12 9939 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)

Theoremfzm1 9940 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzneuz 9941 No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)

Theoremfznuz 9942 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)

Theoremuznfz 9943 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)

Theoremfzp1nel 9944 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.)

Theoremfzrevral 9945* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral2 9946* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral3 9947* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzshftral 9948* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)

Theoremige2m1fz1 9949 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.)

Theoremige2m1fz 9950 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.)

Theoremfz01or 9951 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.)

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: , usually abbreviated by "fz0".

Theoremelfz2nn0 9952 Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfznn0 9953 Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)

Theoremelfznn0 9954 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.)

Theoremelfz3nn0 9955 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.)

Theoremfz0ssnn0 9956 Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)

Theoremfz1ssfz0 9957 Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theorem0elfz 9958 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.)

Theoremnn0fz0 9959 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.)

Theoremelfz0add 9960 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.)

Theoremfz0tp 9961 An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremelfz0ubfz0 9962 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.)

Theoremelfz0fzfz0 9963 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.)

Theoremfz0fzelfz0 9964 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.)

Theoremfznn0sub2 9965 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.)

Theoremuzsubfz0 9966 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.)

Theoremfz0fzdiffz0 9967 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.)

Theoremelfzmlbm 9968 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.)

Theoremelfzmlbp 9969 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.)

Theoremfzctr 9970 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)

Theoremdifelfzle 9971 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.)

Theoremdifelfznle 9972 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.)

Theoremnn0split 9973 Express the set of nonnegative integers as the disjoint (see nn0disj 9975) union of the first values and the rest. (Contributed by AV, 8-Nov-2019.)

Theoremnnsplit 9974 Express the set of positive integers as the disjoint union of the first values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremnn0disj 9975 The first elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)

Theorem1fv 9976 A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theorem4fvwrd4 9977* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)

Theorem2ffzeq 9978* 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.)

4.5.6  Half-open integer ranges

Syntaxcfzo 9979 Syntax for half-open integer ranges.
..^

Definitiondf-fzo 9980* Define a function generating sets of integers using a half-open range. Read ..^ as the integers from up to, but not including, ; contrast with df-fz 9851, which includes . Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10014 with fzsplit 9891, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremfzof 9981 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoel1 9982 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoel2 9983 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoelz 9984 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremfzoval 9985 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzo 9986 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzo2 9987 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzouz 9988 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzodcel 9989 Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
DECID ..^

Theoremfzolb 9990 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with . This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzolb2 9991 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with . This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzole1 9992 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.)
..^

Theoremelfzolt2 9993 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzolt3 9994 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzolt2b 9995 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremelfzolt3b 9996 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzonel 9997 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
..^

Theoremelfzouz2 9998 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.)
..^

Theoremelfzofz 9999 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^

Theoremelfzo3 10000 Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp. , ..^ . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

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