Home | Intuitionistic Logic Explorer Theorem List (p. 98 of 130) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elfz1eq 9701 | Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.) |
Theorem | elfzubelfz 9702 | 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.) |
Theorem | peano2fzr 9703 | A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.) |
Theorem | fzm 9704* | Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.) |
Theorem | fztri3or 9705 | Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.) |
Theorem | fzdcel 9706 | Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.) |
DECID | ||
Theorem | fznlem 9707 | A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.) |
Theorem | fzn 9708 | A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.) |
Theorem | fzen 9709 | A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.) |
Theorem | fz1n 9710 | A 1-based finite set of sequential integers is empty iff it ends at index . (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | 0fz1 9711 | Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | fz10 9712 | There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Theorem | uzsubsubfz 9713 | 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.) |
Theorem | uzsubsubfz1 9714 | 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.) |
Theorem | ige3m2fz 9715 | 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.) |
Theorem | fzsplit2 9716 | Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.) |
Theorem | fzsplit 9717 | Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.) |
Theorem | fzdisj 9718 | Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Theorem | fz01en 9719 | 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.) |
Theorem | elfznn 9720 | A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.) |
Theorem | elfz1end 9721 | A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
Theorem | fz1ssnn 9722 | A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
Theorem | fznn0sub 9723 | 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.) |
Theorem | fzmmmeqm 9724 | 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.) |
Theorem | fzaddel 9725 | Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.) |
Theorem | fzsubel 9726 | Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.) |
Theorem | fzopth 9727 | A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzass4 9728 | Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Theorem | fzss1 9729 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzss2 9730 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Theorem | fzssuz 9731 | A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.) |
Theorem | fzsn 9732 | A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | fzssp1 9733 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzssnn 9734 | Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.) |
Theorem | fzsuc 9735 | 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.) |
Theorem | fzpred 9736 | Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.) |
Theorem | fzpreddisj 9737 | A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.) |
Theorem | elfzp1 9738 | Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzp1ss 9739 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzelp1 9740 | Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzp1elp1 9741 | 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.) |
Theorem | fznatpl1 9742 | Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.) |
Theorem | fzpr 9743 | A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | fztp 9744 | A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzsuc2 9745 | Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzp1disj 9746 | is the disjoint union of with . (Contributed by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzdifsuc 9747 | Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.) |
Theorem | fzprval 9748* | Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.) |
Theorem | fztpval 9749* | Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.) |
Theorem | fzrev 9750 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev2 9751 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev2i 9752 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev3 9753 | The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fzrev3i 9754 | The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fznn 9755 | Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.) |
Theorem | elfz1b 9756 | Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) |
Theorem | elfzm11 9757 | Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | uzsplit 9758 | Express an upper integer set as the disjoint (see uzdisj 9759) union of the first values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Theorem | uzdisj 9759 | The first elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Theorem | fseq1p1m1 9760 | 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.) |
Theorem | fseq1m1p1 9761 | Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | fz1sbc 9762* | Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.) |
Theorem | elfzp1b 9763 | 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.) |
Theorem | elfzm1b 9764 | 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.) |
Theorem | elfzp12 9765 | Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.) |
Theorem | fzm1 9766 | Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | fzneuz 9767 | No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.) |
Theorem | fznuz 9768 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.) |
Theorem | uznfz 9769 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.) |
Theorem | fzp1nel 9770 | 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.) |
Theorem | fzrevral 9771* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrevral2 9772* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrevral3 9773* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fzshftral 9774* | Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.) |
Theorem | ige2m1fz1 9775 | 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.) |
Theorem | ige2m1fz 9776 | 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.) |
Theorem | fz01or 9777 | 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.) |
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". | ||
Theorem | elfz2nn0 9778 | Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fznn0 9779 | Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.) |
Theorem | elfznn0 9780 | 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.) |
Theorem | elfz3nn0 9781 | 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.) |
Theorem | fz0ssnn0 9782 | Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.) |
Theorem | fz1ssfz0 9783 | Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Theorem | 0elfz 9784 | 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.) |
Theorem | nn0fz0 9785 | 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.) |
Theorem | elfz0add 9786 | 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.) |
Theorem | fz0tp 9787 | An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Theorem | elfz0ubfz0 9788 | 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.) |
Theorem | elfz0fzfz0 9789 | 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.) |
Theorem | fz0fzelfz0 9790 | 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.) |
Theorem | fznn0sub2 9791 | 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.) |
Theorem | uzsubfz0 9792 | 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.) |
Theorem | fz0fzdiffz0 9793 | 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.) |
Theorem | elfzmlbm 9794 | 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.) |
Theorem | elfzmlbp 9795 | 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.) |
Theorem | fzctr 9796 | Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.) |
Theorem | difelfzle 9797 | 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.) |
Theorem | difelfznle 9798 | 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.) |
Theorem | nn0split 9799 | Express the set of nonnegative integers as the disjoint (see nn0disj 9801) union of the first values and the rest. (Contributed by AV, 8-Nov-2019.) |
Theorem | nnsplit 9800 | Express the set of positive integers as the disjoint union of the first values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |