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Theorem List for Metamath Proof Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhashunx 11601 The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 11597. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

Theoremhashge0 11602 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Theoremhashgt0 11603 The cardinality of a non-empty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Theoremhashge1 11604 The cardinality of a non-empty set is greater or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)

Theoremhashnn0n0nn 11605 If a nonnegative integer is the size of a set which contains at least one element, this integer is a positive integer. (Contributed by Alexander van der Vekens, 9-Jan-2018.)

Theoremhashunsng 11606 The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremhashprg 11607 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.)

Theoremelprchashprn2 11608 If one element of an unordered pair is not a set, the size of the unordered pair is not 2. (Contributed by Alexander van der Vekens, 7-Oct-2017.)

Theoremhashprb 11609 The size of an unordered pair is 2 if and only if its elements are different sets. (Contributed by Alexander van der Vekens, 17-Jan-2018.)

Theoremhashle00 11610 If the size of a set is less than or equal to zero, the set must be empty. (Contributed by Alexander van der Vekens, 6-Jan-2018.)

Theoremhashgt0elex 11611* If the size of a set is greater than zero, the set must contain at least one element. (Contributed by Alexander van der Vekens, 6-Jan-2018.)

Theoremhashgt0elexb 11612* The size of a set is greater than zero if and only if the set contains at least one element. (Contributed by Alexander van der Vekens, 18-Jan-2018.)

Theoremhashp1i 11613 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash1 11614 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash2 11615 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash3 11616 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash4 11617 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhashssdif 11618 The size of the difference of a finite set and a subset is the set's size minus the subset's. (Contributed by Steve Rodriguez, 24-Oct-2015.)

Theoremhashdif 11619 The size of the difference of a finite set and another set is the first set's size minus that of the intersection of both. (Contributed by Steve Rodriguez, 24-Oct-2015.)

Theoremhashdifsn 11620 The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)

Theoremhashsnlei 11621 Get an upper bound on a concretely specified finite set. Base case: singleton set. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhash1snb 11622* The size of a set is 1 if and only if it is a singleton (containing a set). (Contributed by Alexander van der Vekens, 7-Dec-2017.)

Theoremhashgt12el 11623* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)

Theoremhashgt12el2 11624* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)

Theoremhashunlei 11625 Get an upper bound on a concretely specified finite set. Induction step: union of two finite bounded sets. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashsslei 11626 Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashprlei 11627 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhash2pr 11628* A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremhash2prde 11629* A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremhash2prb 11630* A set of size two is an unordered pair if and only if it contains two different elements. (Contributed by Alexander van der Vekens, 14-Jan-2018.)

Theoremhashtplei 11631 An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashtpg 11632 The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.)

Theoremhashfz 11633 Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)

Theoremfzsdom2 11634 Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.)

Theoremhashfzo 11635 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremhashfzo0 11636 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremhashxplem 11637 Lemma for hashxp 11638. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremhashxp 11638 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremhashmap 11639 The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.)

Theoremhashpw 11640 The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.)

Theoremhashfun 11641 A finite set is a function iff it is equinumerous to its domain. (Contributed by Mario Carneiro, 26-Sep-2013.) (Revised by Mario Carneiro, 12-Mar-2015.)

Theoremhashbclem 11642* Lemma for hashbc 11643: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremhashbc 11643* The binomial coefficient counts the number of subsets of a finite set of a given size. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremhashfacen 11644* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)

Theoremhashf1lem1 11645* Lemma for hashf1 11647. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremhashf1lem2 11646* Lemma for hashf1 11647. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremhashf1 11647* The permutation number counts the number of injections from to . (Contributed by Mario Carneiro, 21-Jan-2015.)

Theoremhashfac 11648* A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.)

Theoremleiso 11649 Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremleisorel 11650 Version of isorel 5999 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)

Theoremfz1isolem 11651* Lemma for fz1iso 11652. (Contributed by Mario Carneiro, 2-Apr-2014.)
OrdIso

Theoremfz1iso 11652* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)

Theoremseqcoll 11653* The function contains a sparse set of non-zero values to be summed. The function is an order isomorphism from the set of non-zero values of to a 1-based finite sequence, and collects these non-zero values together. Under these conditions, the sum over the values in yields the same result as the sum over the original set . (Contributed by Mario Carneiro, 2-Apr-2014.)

Theoremseqcoll2 11654* The function contains a sparse set of non-zero values to be summed. The function is an order isomorphism from the set of non-zero values of to a 1-based finite sequence, and collects these non-zero values together. Under these conditions, the sum over the values in yields the same result as the sum over the original set . (Contributed by Mario Carneiro, 13-Dec-2014.)

5.6.8.1  Finite induction on the size of the first component of a binary relation

Theorembrfi1indlem 11655 Lemma for brfi1ind 11657: The size of a set is the size of this set with one element removed, increased by 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.)

Theorembrfi1uzind 11656* Properties of a binary relation with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, usually with (see brfi1ind 11657) or . (Contributed by Alexander van der Vekens, 7-Jan-2018.)

Theorembrfi1ind 11657* Properties of a binary relation with a finite first component, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, e.g. usgrafis 21309. (Contributed by Alexander van der Vekens, 7-Jan-2018.)

5.6.9  Words over a set

Syntaxcword 11658 Syntax for the Word operator.
Word

Syntaxcconcat 11659 Syntax for the concatenation operator.
concat

Syntaxcs1 11660 Syntax for the singleton word constructor.

Syntaxcsubstr 11661 Syntax for the word slicing operator.
substr

Syntaxcsplice 11662 Syntax for the word splicing operator.
splice

Syntaxcreverse 11663 Syntax for the word reverse operator.
reverse

Definitiondf-word 11664* Define the class of words over a set. A word is a finite sequence of symbols from a set. The domain is forced so that two words with the same symbols in the same order will be the same. This is sometimes denoted with the Kleene star, although properly speaking that is an operator on languages. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Definitiondf-concat 11665* Define the concatenation operator which combines two words. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)
concat ..^ ..^

Definitiondf-s1 11666 Define the canonical injection from symbols to words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Definitiondf-substr 11667* Define an operation which extracts portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
substr ..^ ..^

Definitiondf-splice 11668* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
splice substr concat concat substr

Definitiondf-reverse 11669* Define an operation which reverses the order of symbols in a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse ..^

Theoremiswrd 11670* Property of being a word over a set with a quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Theoremwrdval 11671* Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Theoremiswrdi 11672 A one-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
..^ Word

Theoremwrd0 11673 The empty set is a word (frequently denoted ε in this context). (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word

Theoremwrdf 11674 A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word ..^

Theoremwrdfin 11675 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.)
Word

Theoremlencl 11676 The length of a word is a nonnegative integer. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word

Theoremlennncl 11677 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word

Theoremsswrd 11678 The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word Word

Theoremwrdeq 11679 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word Word

Theoremwrdexg 11680 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theoremnfwrd 11681 Hypothesis builder for Word . (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theoremccatfn 11682 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.)
concat

Theoremccatfval 11683* Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
concat ..^ ..^

Theoremccatcl 11684 The concatenation of two words is a word. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Stefan O'Rear, 15-Aug-2015.)
Word Word concat Word

Theoremccatlen 11685 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word Word concat

Theoremccatval1 11686 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Word Word ..^ concat

Theoremccatval2 11687 Value of a symbol in the right half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Word Word ..^ concat

Theoremccatval3 11688 Value of a symbol in the right half of a concatenated word, using an index relative to the subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Word Word ..^ concat

Theoremccatlid 11689 Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word concat

Theoremccatrid 11690 Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word concat

Theoremccatass 11691 Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word Word Word concat concat concat concat

Theoremids1 11692 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1val 11693 Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems1eq 11694 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1eqd 11695 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems1cl 11696 A singleton word is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1cld 11697 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1cli 11698 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems1len 11699 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems1nz 11700 A singleton is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.)

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