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Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdainflem 7701 Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)

Theoremcdainf 7702 A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfcda1 7703 An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)

Theorempwcda1 7704 The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)

Theorempwcdaidm 7705 If the natural numbers inject into , then is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremcdalepw 7706 If is idempotent under cardinal sum and is dominated by the power set of , then so is the cardinal sum of and . (Contributed by Mario Carneiro, 15-May-2015.)

Theoremonacda 7707 The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)

Theoremcardacda 7708 The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremcdanum 7709 The cardinal sum of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremunnum 7710 The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremnnacda 7711 The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)

Theoremficardun 7712 The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremficardun2 7713 The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)

Theorempwsdompw 7714* Lemma for domtriom 7953. This is the equinumerosity version of the algebraic identity . (Contributed by Mario Carneiro, 7-Feb-2013.)

Theoremunctb 7715 The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)

Theoreminfcdaabs 7716 Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfunabs 7717 An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfcda 7718 The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfdif 7719 The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfdif2 7720 Cardinality ordering for an infinite set difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpdom 7721 Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpabs 7722 Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfunsdom1 7723 The union of two sets that are strictly dominated by the infinite set is also dominated by . This version of infunsdom 7724 assumes additionally that is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theoreminfunsdom 7724 The union of two sets that are strictly dominated by the infinite set is also strictly dominated by . (Contributed by Mario Carneiro, 3-May-2015.)

Theoreminfxp 7725 Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theorempwcdadom 7726 A property of dominance over a powerset, and a main lemma for gchac 8175. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)

Theoreminfpss 7727* Every infinite set has an equinumerous proper subset. Exercise 7 of [TakeutiZaring] p. 91. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoreminfmap2 7728* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8078 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

2.6.10  The Ackermann bijection

Theoremackbij2lem1 7729 Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem1 7730 Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem2 7731 Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem3 7732 Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem4 7733 Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Theoremackbij1lem5 7734 Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Theoremackbij1lem6 7735 Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem7 7736* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremackbij1lem8 7737* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Theoremackbij1lem9 7738* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Theoremackbij1lem10 7739* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem11 7740* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem12 7741* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem13 7742* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem14 7743* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem15 7744* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem16 7745* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem17 7746* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1lem18 7747* Lemma for ackbij1 7748. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1 7748* The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij1b 7749* The Ackermann bijection, part 1b: the bijection from ackbij1 7748 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij2lem2 7750* Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij2lem3 7751* Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij2lem4 7752* Lemma for ackbij2 7753. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremackbij2 7753* The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremr1om 7754 The set of hereditarily finite sets is countable. See ackbij2 7753 for an explicit bijection that works without Infinity. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremfictb 7755 A set is countable iff its collection of finite intersections is countable. (Contributed by Jeff Hankins, 24-Aug-2009.) (Proof shortened by Mario Carneiro, 17-May-2015.)

2.6.11  Cofinality (without Axiom of Choice)

Theoremcflem 7756* A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set . (Contributed by NM, 24-Apr-2004.)

Theoremcfval 7757* Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number is the cardinality (size) of the smallest unbounded subset of the ordinal number. Unbounded means that for every member of , there is a member of that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremcff 7758 Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theoremcfub 7759* An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremcflm 7760* Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)

Theoremcf0 7761 Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)

Theoremcardcf 7762 Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremcflecard 7763 Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremcfle 7764 Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremcfon 7765 The cofinality of any set is an ordinal (although it only makes sense when is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoremcfeq0 7766 Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)

Theoremcfsuc 7767 Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)

Theoremcff1 7768* There is always a map from to (this is a stronger condition than the definition, which only presupposes a map from some . (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcfflb 7769* If there is a cofinal map from to , then is at least . This theorem and cff1 7768 motivate the picture of as the greatest lower bound of the domain of cofinal maps into . (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcfval2 7770* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcoflim 7771* A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcflim3 7772* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcflim2 7773 The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremcfom 7774 Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Proof shortened by Mario Carneiro, 11-Jun-2015.)

Theoremcfss 7775* There is a cofinal subset of of cardinality . (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremcfslb 7776 Any cofinal subset of is at least as large as . (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremcfslbn 7777 Any subset of smaller than its cofinality has union less than . (This is the contrapositive to cfslb 7776.) (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremcfslb2n 7778* Any small collection of small subsets of cannot have union , where "small" means smaller than the cofinality. This is a stronger version of cfslb 7776. This is a common application of cofinality: under AC, is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremcofsmo 7779* Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by by Mario Carneiro, 20-Mar-2013.)
OrdIso

Theoremcfsmolem 7780* Lemma for cfsmo 7781. (Contributed by Mario Carneiro, 28-Feb-2013.)
recs

Theoremcfsmo 7781* The map in cff1 7768 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcfcoflem 7782* Lemma for cfcof 7784, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremcoftr 7783* If there is a cofinal map from to and another from to , then there is also a cofinal map from to . Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 7784. (Contributed by Mario Carneiro, 16-Mar-2013.)

Theoremcfcof 7784* If there is a cofinal map from to , then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof and defines our as the minimum such that cof . (Contributed by Mario Carneiro, 20-Mar-2013.)

Theoremcfidm 7785 The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremalephsing 7786 The cofinality of a limit aleph is the same as the cofinality of its argument, so if , then is singular. Conversely, if is regular (i.e. weakly inaccessible), then , so has to be rather large (see alephfp 7619). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)

2.6.12  Eight inequivalent definitions of finite set

Theoremsornom 7787* The range of a single-step monotone function from into a partially ordered set is a chain. (Contributed by Stefan O'Rear, 3-Nov-2014.)

Syntaxcfin1a 7788 Extend class notation to include the class of Ia-finite sets.
FinIa

Syntaxcfin2 7789 Extend class notation to include the class of II-finite sets.
FinII

Syntaxcfin4 7790 Extend class notation to include the class of IV-finite sets.
FinIV

Syntaxcfin3 7791 Extend class notation to include the class of III-finite sets.
FinIII

Syntaxcfin5 7792 Extend class notation to include the class of V-finite sets.
FinV

Syntaxcfin6 7793 Extend class notation to include the class of VI-finite sets.
FinVI

Syntaxcfin7 7794 Extend class notation to include the class of VII-finite sets.
FinVII

Definitiondf-fin1a 7795* A set is Ia-finite iff it is not the union of two I-infinite sets. Equivalent to definition Ia of [Levy58] p. 2. A I-infinite Ia-finite set is also known as an amorphous set. This is the second of Levy's eight definitions of finite set. Levy's I-finite is equivalent to our df-fin 6753 and not repeated here. These eight definitions are equivalent with Choice but strictly decreasing in strength in models where Choice fails; conversely, they provide a series of increasingly stronger notions of infiniteness. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinIa

Definitiondf-fin2 7796* A set is II-finite (Tarski finite) iff every nonempty chain of subsets contains a maximum element. Definition II of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinII []

Definitiondf-fin4 7797* A set is IV-finite (Dedekind finite) iff it has no equinumerous proper subset. Definition IV of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinIV

Definitiondf-fin3 7798 A set is III-finite (weakly Dedekind finite) iff its power set is Dedekind finite. Definition III of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinIII FinIV

Definitiondf-fin5 7799 A set is V-finite iff it behaves finitely under . Definition V of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinV

Definitiondf-fin6 7800 A set is VI-finite iff it behaves finitely under . Definition VI of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinVI

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