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Theorem List for Metamath Proof Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisfinite2 7401 Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity. (Contributed by NM, 24-Apr-2004.)

Theoremnnsdomg 7402 Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 15-Jun-1998.)

Theoremisfiniteg 7403 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoreminfsdomnn 7404 An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoreminfn0 7405 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)

Theoremfin2inf 7406 This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless exists. (Contributed by NM, 13-Nov-2003.)

Theoremunfilem1 7407* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremunfilem2 7408* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremunfilem3 7409 Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 16-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremunfi 7410 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.)

Theoremunfir 7411 If a union is finite, the operands are finite. Converse of unfi 7410. (Contributed by FL, 3-Aug-2009.)

Theoremunfi2 7412 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 7410 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7406). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoremdifinf 7413 An infinite set minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)

Theoremxpfi 7414 The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)

Theoremdomunfican 7415 A finite set union cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)

Theoreminfcntss 7416* Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)

Theoremprfi 7417 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)

Theoremtpfi 7418 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)

Theoremfiint 7419* Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of is in ." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally. (Contributed by NM, 22-Sep-2002.)

Theoremfnfi 7420 A version of fnex 5997 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremfodomfi 7421 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8440 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)

Theoremfodomfib 7422* Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 8442 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)

Theoremfofinf1o 7423 Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)

Theoremfidomdm 7424 Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremdmfi 7425 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)

Theoremcnvfi 7426 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremrnfi 7427 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremfofi 7428 If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.)

Theoremf1fi 7429 If a 1-to-1 function has a finite codomain its domain is finite. (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremiunfi 7430* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 7431. Note that depends on , i.e. can be thought of as . (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremunifi 7431 The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. (Contributed by NM, 22-Aug-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremunifi2 7432* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 7431 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7406). (Contributed by NM, 11-Mar-2006.)

Theoreminfssuni 7433* If an infinite set is included in the underlying set of a finite cover , then there exists a set of the cover that contains an infinite number of element of . (Contributed by FL, 2-Aug-2009.)

Theoremunirnffid 7434 The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremimafi 7435 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremsuppfif1 7436 Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theorempwfilem 7437* Lemma for pwfi 7438. (Contributed by NM, 26-Mar-2007.)

Theorempwfi 7438 The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.)

Theoremmapfi 7439 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremixpfi 7440* A cross product of finitely many finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremixpfi2 7441* A cross product of finite sets such that all but finitely many are singletons is finite. (Note that and are both possibly dependent on . ) (Contributed by Mario Carneiro, 25-Jan-2015.)

Theoremmptfi 7442* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremabrexfi 7443* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)

Theoremelfpw 7444 Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremunifpw 7445 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Theoremf1opwfi 7446* A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)

Theoremfissuni 7447* A finite subset of a union is covered by finitely many elements. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Theoremfipreima 7448* Given a finite subset of the range of a function, there exists a finite subset of the domain whose image is . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 22-Feb-2015.)

Theoremfinsschain 7449* A finite subset of the union of a superset chain is a subset of some element of the chain. A useful preliminary result for alexsub 18114 and others. (Contributed by Jeff Hankins, 25-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 18-May-2015.)
[]

Theoremindexfi 7450* If for every element of a finite indexing set there exists a corresponding element of another set , then there exists a finite subset of consisting only of those elements which are indexed by . Proven without the Axiom of Choice, unlike indexdom 26478. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

2.4.36  Finite intersections

Syntaxcfi 7451 Extend class notation with the function whose value is the class of all the finite intersections of the elements of a given set.

Definitiondf-fi 7452* Function whose value is the class of all the finite intersections of the elements of . (Contributed by FL, 27-Apr-2008.)

Theoremfival 7453* The set of all the finite intersections of the elements of . (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremelfi 7454* Specific properties of an element of . (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremelfi2 7455* The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremelfir 7456 Sufficient condition for an element of . (Contributed by Mario Carneiro, 24-Nov-2013.)

Theoremintrnfi 7457 Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremiinfi 7458* An indexed intersection of elements of is an element of the finite intersections of . (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoreminelfi 7459 The interesection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)

Theoremssfii 7460 Any element of a set is the intersection of a finite subset of . (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfi0 7461 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremfieq0 7462 If is not empty, the class of all the finite intersections of is not empty either. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremfiin 7463 The elements of are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)

Theoremdffi2 7464* The set of finite intersections is the smallest set that contains and is closed under pairwise intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)

Theoremfiss 7465 Subset relationship for function . (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoreminficl 7466* A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremfipwuni 7467 The set of finite intersections of a set is contained in the powerset of the union of the elements of . (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfisn 7468 A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremfiuni 7469 The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremfipwss 7470 If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremelfiun 7471* A finite intersection of elements taken from a union of collections. (Contributed by Jeff Hankins, 15-Nov-2009.) (Proof shortened by Mario Carneiro, 26-Nov-2013.)

Theoremdffi3 7472* The set of finite intersections can be "constructed" inductively by iterating binary intersection -many times. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfifo 7473* Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)

2.4.37  Hall's marriage theorem

Theoremmarypha1lem 7474* Core induction for Philip Hall's marriage theorem. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremmarypha1 7475* (Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pidgeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremmarypha2lem1 7476* Lemma for marypha2 7480. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremmarypha2lem2 7477* Lemma for marypha2 7480. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremmarypha2lem3 7478* Lemma for marypha2 7480. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremmarypha2lem4 7479* Lemma for marypha2 7480. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremmarypha2 7480* Version of marypha1 7475 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

2.4.38  Supremum

Syntaxcsup 7481 Extend class notation to include supremum of class . Here is ordinarily a relation that strictly orders class . For example, could be 'less than' and could be the set of real numbers.

Definitiondf-sup 7482* Define the supremum of class . It is meaningful when is a relation that strictly orders and when the supremum exists. For example, could be 'less than', could be the set of real numbers, and could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrval 12080. See dfsup2 7483 for alternate definition not requiring dummy variables.

We will also use this notation for "infimum" by replacing with . (Contributed by NM, 22-May-1999.)

Theoremdfsup2 7483 Quantifier free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdfsup2OLD 7484 Quantifier-free definition of supremum. (Contributed by Scott Fenton, 18-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfsup3OLD 7485 Quantifier-free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsupeq1 7486 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)

Theoremsupeq1d 7487 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremsupeq1i 7488 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremsupeq2 7489 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsupeq3 7490 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)

Theoremsupeq123d 7491 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremnfsup 7492 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)

Theoremsupmo 7493* Any class has at most one supremum in (where is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsupexd 7494 A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsupeu 7495* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)

Theoremsupval2 7496* Alternative expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremeqsup 7497* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremeqsupd 7498* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremsupcl 7499* A supremum belongs to its base class (closure law). See also supub 7500 and suplub 7501. (Contributed by NM, 12-Oct-2004.)

Theoremsupub 7500* A supremum is an upper bound. See also supcl 7499 and suplub 7501.

This proof demonstrates how to expand an iota-based definition (df-iota 5453) using riotacl2 6599.

(Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

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