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Theorem List for Metamath Proof Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremenp1ilem 7301 Lemma for uses of enp1i 7302. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremenp1i 7302* Proof induction for en2i 7104 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremen2 7303* A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremen3 7304* A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremen4 7305* A set equinumerous to ordinal 4 is a quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremfindcard 7306* Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfindcard2 7307* Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)

Theoremfindcard2s 7308* Variation of findcard2 7307 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremfindcard3 7309* Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013.)

Theoremac6sfi 7310* A version of ac6s 8320 for finite sets. (Contributed by Jeffrey Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)

Theoremfrfi 7311 A partial order is well-founded on a finite set. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)

Theoremfimax2g 7312* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)

Theoremfimaxg 7313* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)

Theoremfisupg 7314* Lemma showing existence and closure of supremum of a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremwofi 7315 A total order on a finite set is a well-order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)

Theoremordunifi 7316 The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)

Theoremnnunifi 7317 The union (supremum) of a finite set of finite ordinals is a finite ordinal. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremunblem1 7318* Lemma for unbnn 7322. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set. (Contributed by NM, 3-Dec-2003.)

Theoremunblem2 7319* Lemma for unbnn 7322. The value of the function belongs to the unbounded set of natural numbers . (Contributed by NM, 3-Dec-2003.)

Theoremunblem3 7320* Lemma for unbnn 7322. The value of the function is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)

Theoremunblem4 7321* Lemma for unbnn 7322. The function maps the set of natural numbers one-to-one to the set of unbounded natural numbers . (Contributed by NM, 3-Dec-2003.)

Theoremunbnn 7322* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 7569 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)

Theoremunbnn2 7323* Version of unbnn 7322 that does not require a strict upper bound. (Contributed by NM, 24-Apr-2004.)

Theoremisfinite2 7324 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 7325 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 7326 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 7327 An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoreminfn0 7328 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)

Theoremfin2inf 7329 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 7330* 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 7331* 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 7332 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 7333 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.)

Theoremunfir 7334 If a union is finite, the operands are finite. Converse of unfi 7333. (Contributed by FL, 3-Aug-2009.)

Theoremunfi2 7335 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 7333 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7329). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoremdifinf 7336 An infinite set minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)

Theoremxpfi 7337 The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)

Theoremdomunfican 7338 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 7339* 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 7340 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)

Theoremtpfi 7341 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)

Theoremfiint 7342* 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 7343 A version of fnex 5920 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremfodomfi 7344 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8358 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 7345* Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 8360 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)

Theoremfofinf1o 7346 Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)

Theoremfidomdm 7347 Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremdmfi 7348 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)

Theoremcnvfi 7349 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremrnfi 7350 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremfofi 7351 If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.)

Theoremf1fi 7352 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 7353* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 7354. 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 7354 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 7355* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 7354 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7329). (Contributed by NM, 11-Mar-2006.)

Theoremunirnffid 7356 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 7357 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremsuppfif1 7358 Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theorempwfilem 7359* Lemma for pwfi 7360. (Contributed by NM, 26-Mar-2007.)

Theorempwfi 7360 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 7361 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremixpfi 7362* A cross product of finitely many finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremixpfi2 7363* 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 7364* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremabrexfi 7365* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)

Theoremelfpw 7366 Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremunifpw 7367 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Theoremf1opwfi 7368* A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)

Theoremfissuni 7369* A finite subset of a union is covered by finitely many elements. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Theoremfipreima 7370* 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 7371* 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 18029 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 7372* 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 26326. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

2.4.36  Finite intersections

Syntaxcfi 7373 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 7374* Function whose value is the class of all the finite intersections of the elements of . (Contributed by FL, 27-Apr-2008.)

Theoremfival 7375* 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 7376* Specific properties of an element of . (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremelfi2 7377* The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremelfir 7378 Sufficient condition for an element of . (Contributed by Mario Carneiro, 24-Nov-2013.)

Theoremintrnfi 7379 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 7380* An indexed intersection of elements of is an element of the finite intersections of . (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoreminelfi 7381 The interesection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)

Theoremssfii 7382 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 7383 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremfieq0 7384 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 7385 The elements of are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)

Theoremdffi2 7386* 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 7387 Subset relationship for function . (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoreminficl 7388* 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 7389 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 7390 A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremfiuni 7391 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 7392 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 7393* 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 7394* The set of finite intersections can be "constructed" inductively by iterating binary intersection -many times. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfifo 7395* 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 7396* Core induction for Philip Hall's marriage theorem. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremmarypha1 7397* (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 7398* Lemma for marypha2 7402. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremmarypha2lem2 7399* Lemma for marypha2 7402. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremmarypha2lem3 7400* Lemma for marypha2 7402. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

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