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Type | Label | Description |
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Statement | ||
Theorem | fzrev3i 10101 | The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
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Theorem | fznn 10102 | Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.) |
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Theorem | elfz1b 10103 | Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) |
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Theorem | elfzm11 10104 | Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | uzsplit 10105 |
Express an upper integer set as the disjoint (see uzdisj 10106) union of
the first ![]() |
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Theorem | uzdisj 10106 |
The first ![]() |
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Theorem | fseq1p1m1 10107 | 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.) |
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Theorem | fseq1m1p1 10108 | Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) |
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Theorem | fz1sbc 10109* | Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.) |
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Theorem | elfzp1b 10110 | 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.) |
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Theorem | elfzm1b 10111 | 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.) |
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Theorem | elfzp12 10112 | Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.) |
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Theorem | fzm1 10113 | Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | fzneuz 10114 | No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.) |
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Theorem | fznuz 10115 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.) |
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Theorem | uznfz 10116 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.) |
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Theorem | fzp1nel 10117 | 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.) |
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Theorem | fzrevral 10118* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
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Theorem | fzrevral2 10119* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
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Theorem | fzrevral3 10120* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
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Theorem | fzshftral 10121* | Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.) |
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Theorem | ige2m1fz1 10122 | 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.) |
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Theorem | ige2m1fz 10123 | 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.) |
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Theorem | fz01or 10124 | 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.) |
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Finite intervals of nonnegative integers (or "finite sets of sequential
nonnegative integers") are finite intervals of integers with 0 as lower
bound:
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Theorem | elfz2nn0 10125 | Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
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Theorem | fznn0 10126 | Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.) |
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Theorem | elfznn0 10127 | 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.) |
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Theorem | elfz3nn0 10128 | 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.) |
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Theorem | fz0ssnn0 10129 | Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.) |
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Theorem | fz1ssfz0 10130 | Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | 0elfz 10131 | 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.) |
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Theorem | nn0fz0 10132 | 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.) |
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Theorem | elfz0add 10133 | 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.) |
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Theorem | fz0sn 10134 | An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.) |
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Theorem | fz0tp 10135 | An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
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Theorem | fz0to3un2pr 10136 | An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021.) |
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Theorem | fz0to4untppr 10137 | 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.) |
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Theorem | elfz0ubfz0 10138 | 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.) |
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Theorem | elfz0fzfz0 10139 | 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.) |
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Theorem | fz0fzelfz0 10140 | 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.) |
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Theorem | fznn0sub2 10141 | 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.) |
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Theorem | uzsubfz0 10142 | 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.) |
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Theorem | fz0fzdiffz0 10143 | 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.) |
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Theorem | elfzmlbm 10144 | 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.) |
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Theorem | elfzmlbp 10145 | 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.) |
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Theorem | fzctr 10146 | Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.) |
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Theorem | difelfzle 10147 | 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.) |
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Theorem | difelfznle 10148 | 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.) |
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Theorem | nn0split 10149 |
Express the set of nonnegative integers as the disjoint (see nn0disj 10151)
union of the first ![]() ![]() ![]() |
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Theorem | nnsplit 10150 |
Express the set of positive integers as the disjoint union of the first
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Theorem | nn0disj 10151 |
The first ![]() ![]() ![]() |
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Theorem | 1fv 10152 | A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
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Theorem | 4fvwrd4 10153* | The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.) |
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Theorem | 2ffzeq 10154* | 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.) |
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Syntax | cfzo 10155 | Syntax for half-open integer ranges. |
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Definition | df-fzo 10156* |
Define a function generating sets of integers using a half-open range.
Read ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fzof 10157 | Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
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Theorem | elfzoel1 10158 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
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Theorem | elfzoel2 10159 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
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Theorem | elfzoelz 10160 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
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Theorem | fzoval 10161 | Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
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Theorem | elfzo 10162 | Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
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Theorem | elfzo2 10163 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
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Theorem | elfzouz 10164 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
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Theorem | fzodcel 10165 | Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.) |
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Theorem | fzolb 10166 |
The left endpoint of a half-open integer interval is in the set iff the
two arguments are integers with ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fzolb2 10167 |
The left endpoint of a half-open integer interval is in the set iff the
two arguments are integers with ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | elfzole1 10168 | 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.) |
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Theorem | elfzolt2 10169 | A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
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Theorem | elfzolt3 10170 | Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
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Theorem | elfzolt2b 10171 | A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.) |
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Theorem | elfzolt3b 10172 | Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.) |
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Theorem | fzonel 10173 | A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
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Theorem | elfzouz2 10174 | 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.) |
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Theorem | elfzofz 10175 | A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
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Theorem | elfzo3 10176 |
Express membership in a half-open integer interval in terms of the "less
than or equal" and "less than" predicates on integers,
resp.
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Theorem | fzom 10177* | A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.) |
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Theorem | fzossfz 10178 | 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.) |
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Theorem | fzon 10179 | 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.) |
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Theorem | fzonlt0 10180 | A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.) |
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Theorem | fzo0 10181 | Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
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Theorem | fzonnsub 10182 |
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Theorem | fzonnsub2 10183 |
If ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fzoss1 10184 | Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
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Theorem | fzoss2 10185 | Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
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Theorem | fzossrbm1 10186 | Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.) |
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Theorem | fzo0ss1 10187 | Subset relationship for half-open integer ranges with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
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Theorem | fzossnn0 10188 | 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.) |
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Theorem | fzospliti 10189 | One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
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Theorem | fzosplit 10190 | Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
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Theorem | fzodisj 10191 | Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
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Theorem | fzouzsplit 10192 | Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.) |
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Theorem | fzouzdisj 10193 | 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.) |
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Theorem | lbfzo0 10194 |
An integer is strictly greater than zero iff it is a member of ![]() |
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Theorem | elfzo0 10195 | Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
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Theorem | fzo1fzo0n0 10196 | An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.) |
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Theorem | elfzo0z 10197 | Membership in a half-open range of nonnegative integers, generalization of elfzo0 10195 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
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Theorem | elfzo0le 10198 | A member in a half-open range of nonnegative integers is less than or equal to the upper bound of the range. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
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Theorem | elfzonn0 10199 | A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.) |
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Theorem | fzonmapblen 10200 | The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one is less then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.) |
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