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

Theorembcrpcl 11601 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 11616.) (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembccmpl 11602 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)

Theorembcn0 11603 choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembc0k 11604 The binomial coefficient " 0 choose " is 0 for a positive integer K. Note that (see bcn0 11603). (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Theorembcnn 11605 choose is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcn1 11606 Binomial coefficient: choose . (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcnp1n 11607 Binomial coefficient: choose . (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcm1k 11608 The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1n 11609 The proportion of one binomial coefficient to another with increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1nk 11610 The proportion of one binomial coefficient to another with and increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorembcval5 11611 Write out the top and bottom parts of the binomial coefficient explicitly. In this form, it is valid even for , although it is no longer valid for non-positive . (Contributed by Mario Carneiro, 22-May-2014.)

Theorembcn2 11612 Binomial coefficient: choose . (Contributed by Mario Carneiro, 22-May-2014.)

Theorembcp1m1 11613 Compute the binomial coefficent of over (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)

Theorembcpasc 11614 Pascal's rule for the binomial coefficient, generalized to all integers . Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)

Theorembccl 11615 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theorembccl2 11616 A binomial coefficient, in its standard domain, is a natural number. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)

Theorembcn2m1 11617 Compute the binomial coefficient " choose 2 " from " choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ) (Contributed by Alexander van der Vekens, 7-Jan-2018.)

Theorembcn2p1 11618 Compute the binomial coefficient " choose 2 " from " choose 2 ": N + ( N 2 ) = ( (N+1) 2 ) (Contributed by Alexander van der Vekens, 8-Jan-2018.)

Theorempermnn 11619 The number of permutations of objects from a collection of objects is a natural number. (Contributed by Jason Orendorff, 24-Jan-2007.)

5.6.8  The ` # ` (finite set size) function

Syntaxchash 11620 Extend the definition of a class to include the size function.

Definitiondf-hash 11621 Define the function, which gives the cardinality of a finite set as a member of , and assigns all infinite sets the value . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremhashkf 11622 The finite part of the size function maps all finite sets to their cardinality, as members of . (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremhashgval 11623* The value of the function in terms of the mapping from to . The proof avoids the use of ax-ac 8341. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremhashginv 11624* maps the size function's value to . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashinf 11625 The value of the function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremhashbnd 11626 If has size bounded by an integer , then is finite. (Contributed by Mario Carneiro, 14-Jun-2015.)

Theoremhashf 11627 The size function maps all finite sets to their cardinality, as members of , and infinite sets to . (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremhashnn0pnf 11628 The value of the hash function for a set is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 6-Dec-2017.)

Theoremhashnnn0genn0 11629 If the size of a set is not a nonnegative integer, it is greater than or equal to any nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.)

Theoremhashnemnf 11630 The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

Theoremhashv01gt1 11631 The size of a set is either 0 or 1 or greater than 1. (Contributed by Alexander van der Vekens, 29-Dec-2017.)

Theoremhashfz1 11632 The set has elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashen 11633 Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhasheni 11634 Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.)

Theoremhasheqf1o 11635* The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)

Theoremfiinfnf1o 11636* There is no bijection between a finite set and an infinite set. (Contributed by Alexander van der Vekens, 25-Dec-2017.)

Theoremhasheqf1oi 11637* The size of two sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 25-Dec-2017.)

Theoremhashf1rn 11638 The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by Alexander van der Vekens, 4-Feb-2018.)

Theoremfz1eqb 11639 Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)

Theoremhashcard 11640 The size function of the cardinality function. (Contributed by Mario Carneiro, 19-Sep-2013.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremhashcl 11641 Closure of the function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremhashxrcl 11642 Extended real closure of the function. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremhashclb 11643 Reverse closure of the function. (Contributed by Mario Carneiro, 15-Jan-2015.)

Theoremhashvnfin 11644 A set of finite size is a finite set. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremhashnfinnn0 11645 The size of an infinite set is not a nonnegative integer. (Contributed by Alexander van der Vekens, 21-Dec-2017.) (Proof shortened by Alexander van der Vekens, 18-Jan-2018.)

Theoremhasheq0 11646 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.)

Theoremhashnncl 11647 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theoremhash0 11648 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)

Theoremhashsng 11649 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

Theoremhashrabrsn 11650* The size of a restricted class abstraction restricted to a singleton is a nonnegative integer. (Contributed by Alexander van der Vekens, 22-Dec-2017.)

Theoremhashfn 11651 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)

Theoremfseq1hash 11652 The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)

Theoremhashgadd 11653 maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashgval2 11654 A short expression for the function of hashgf1o 11312. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremhashdom 11655 Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)

Theoremhashdomi 11656 Non-strict order relation of the function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theoremhashsdom 11657 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremhashun 11658 The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashun2 11659 The size of the union of finite sets is less than or equal to the sum of their sizes. (Contributed by Mario Carneiro, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 27-Jul-2014.)

Theoremhashun3 11660 The size of the union of finite sets is the sum of their sizes minus the size of the intersection. (Contributed by Mario Carneiro, 6-Aug-2017.)

Theoremhashinfxadd 11661 The extended real addition of the size of an infinite set with the size of an arbitrary set yields plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)

Theoremhashunx 11662 The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 11658. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

Theoremhashge0 11663 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Theoremhashgt0 11664 The cardinality of a non-empty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Theoremhashge1 11665 The cardinality of a non-empty set is greater or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)

Theoremhashnn0n0nn 11666 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 11667 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 11668 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.)

Theoremelprchashprn2 11669 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 11670 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 11671 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 11672* 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 11673* 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 11674 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash1 11675 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash2 11676 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash3 11677 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash4 11678 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhashssdif 11679 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 11680 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 11681 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 11682 Get an upper bound on a concretely specified finite set. Base case: singleton set. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhash1snb 11683* 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 11684* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)

Theoremhashgt12el2 11685* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)

Theoremhashunlei 11686 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 11687 Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashprlei 11688 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhash2pr 11689* A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremhash2prde 11690* A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremhash2prb 11691* 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 11692 An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashtpg 11693 The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.)

Theoremhashfz 11694 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 11695 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 11696 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremhashfzo0 11697 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremhashxplem 11698 Lemma for hashxp 11699. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremhashxp 11699 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremhashmap 11700 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.)

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