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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcfss 8101* There is a cofinal subset of of cardinality . (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremcfslb 8102 Any cofinal subset of is at least as large as . (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremcfslbn 8103 Any subset of smaller than its cofinality has union less than . (This is the contrapositive to cfslb 8102.) (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremcfslb2n 8104* Any small collection of small subsets of cannot have union , where "small" means smaller than the cofinality. This is a stronger version of cfslb 8102. 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 8105* 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 Mario Carneiro, 20-Mar-2013.)
OrdIso

Theoremcfsmolem 8106* Lemma for cfsmo 8107. (Contributed by Mario Carneiro, 28-Feb-2013.)
recs

Theoremcfsmo 8107* The map in cff1 8094 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremcfcoflem 8108* Lemma for cfcof 8110, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremcoftr 8109* 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 8110. (Contributed by Mario Carneiro, 16-Mar-2013.)

Theoremcfcof 8110* 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 8111 The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremalephsing 8112 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 7945). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)

2.6.12  Eight inequivalent definitions of finite set

Theoremsornom 8113* 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 8114 Extend class notation to include the class of Ia-finite sets.
FinIa

Syntaxcfin2 8115 Extend class notation to include the class of II-finite sets.
FinII

Syntaxcfin4 8116 Extend class notation to include the class of IV-finite sets.
FinIV

Syntaxcfin3 8117 Extend class notation to include the class of III-finite sets.
FinIII

Syntaxcfin5 8118 Extend class notation to include the class of V-finite sets.
FinV

Syntaxcfin6 8119 Extend class notation to include the class of VI-finite sets.
FinVI

Syntaxcfin7 8120 Extend class notation to include the class of VII-finite sets.
FinVII

Definitiondf-fin1a 8121* 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 7072 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 8122* 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 8123* 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 8124 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 8125 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 8126 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

Definitiondf-fin7 8127* A set is VII-finite iff it cannot be infinitely well-ordered. Equivalent to definition VII of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinVII

Theoremisfin1a 8128* Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIa

Theoremfin1ai 8129 Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIa

Theoremisfin2 8130* Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinII []

Theoremfin2i 8131 Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinII []

Theoremisfin3 8132 Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIII FinIV

Theoremisfin4 8133* Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIV

Theoremfin4i 8134 Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIV

Theoremisfin5 8135 Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinV

Theoremisfin6 8136 Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinVI

Theoremisfin7 8137* Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinVII

Theoremsdom2en01 8138 A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem1 8139 Lemma for infpssr 8144. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem2 8140 Lemma for infpssr 8144. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem3 8141 Lemma for infpssr 8144. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem4 8142 Lemma for infpssr 8144. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem5 8143 Lemma for infpssr 8144. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssr 8144 Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Theoremfin4en1 8145 Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
FinIV FinIV

Theoremssfin4 8146 Dedekind finite sets have Dedekind finite subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Revised by Mario Carneiro, 6-May-2015.)
FinIV FinIV

Theoremdomfin4 8147 A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)
FinIV FinIV

Theoremominf4 8148 is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
FinIV

TheoreminfpssALT 8149* A set with a denumerable subset has a proper subset equinumerous to it, proved without AC or Infinity. Unlike infpss 8053, it uses Replacement. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremisfin4-2 8150 Alternate definition of IV-finite sets: they lack a denumerable subset. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
FinIV

Theoremisfin4-3 8151 Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 8133 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
FinIV

Theoremfin23lem7 8152* Lemma for isfin2-2 8155. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Theoremfin23lem11 8153* Lemma for isfin2-2 8155. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Theoremfin2i2 8154 A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)
FinII []

Theoremisfin2-2 8155* FinII expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
FinII []

Theoremssfin2 8156 A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.)
FinII FinII

Theoremenfin2i 8157 II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
FinII FinII

Theoremfin23lem24 8158 Lemma for fin23 8225. In a class of ordinals, each element is fully identified by those of its predecessors which also belong to the class. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfincssdom 8159 In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.)

Theoremfin23lem25 8160 Lemma for fin23 8225. In a chain of finite sets, equinumerousity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfin23lem26 8161* Lemma for fin23lem22 8163. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfin23lem23 8162* Lemma for fin23lem22 8163. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfin23lem22 8163* Lemma for fin23 8225 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 8164) between an infinite subset of and itself. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfin23lem27 8164* The mapping constructed in fin23lem22 8163 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremisfin3ds 8165* Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.)

Theoremssfin3ds 8166* A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014.)

Theoremfin23lem12 8167* The beginning of the proof that every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets).

This first section is dedicated to the construction of and its intersection. First, the value of at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)

seq𝜔

Theoremfin23lem13 8168* Lemma for fin23 8225. Each step of is a decrease. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem14 8169* Lemma for fin23 8225. will never evolve to an empty set if it did not start with one. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem15 8170* Lemma for fin23 8225. is a monotone function. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem16 8171* Lemma for fin23 8225. ranges over the original set; in particular is a set, although we do not assume here that is. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem19 8172* Lemma for fin23 8225. The first set in to see an input set is either contained in it or disjoint from it. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem20 8173* Lemma for fin23 8225. is either contained in or disjoint from all input sets. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem17 8174* Lemma for fin23 8225. By ? Fin3DS ? , achieves its minimum ( in the synopsis above, but we will not be assigning a symbol here). TODO: Fix comment; math symbol Fin3DS does not exist. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
seq𝜔

Theoremfin23lem21 8175* Lemma for fin23 8225. is not empty. We only need here that has at least one set in its range besides ; the much stronger hypothesis here will serve as our induction hypothesis though. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 6-May-2015.)
seq𝜔

Theoremfin23lem28 8176* Lemma for fin23 8225. The residual is also one-to-one. This preserves the induction invariant. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem29 8177* Lemma for fin23 8225. The residual is built from the same elements as the previous sequence. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem30 8178* Lemma for fin23 8225. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem31 8179* Lemma for fin23 8225. The residual is has a strictly smaller range than the previous sequence. This will be iterated to build an unbounded chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem32 8180* Lemma for fin23 8225. Wrap the previous construction into a function to hide the hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem33 8181* Lemma for fin23 8225. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremfin23lem34 8182* Lemma for fin23 8225. Establish induction invariants on which parameterizes our contradictory chain of subsets. In this section, is the hypothetically assumed family of subsets, is the ground set, and is the induction function constructed in the previous section. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremfin23lem35 8183* Lemma for fin23 8225. Strict order property of . (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremfin23lem36 8184* Lemma for fin23 8225. Weak order property of . (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremfin23lem38 8185* Lemma for fin23 8225. The contradictory chain has no minimum. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Theoremfin23lem39 8186* Lemma for fin23 8225. Thus, we have that could not have been in after all. (Contributed by Stefan O'Rear, 4-Nov-2014.)

Theoremfin23lem40 8187* Lemma for fin23 8225. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)
FinII

Theoremfin23lem41 8188* Lemma for fin23 8225. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.)
FinIII

Theoremisf32lem1 8189* Lemma for isfin3-2 8203. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem2 8190* Lemma for isfin3-2 8203. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem3 8191* Lemma for isfin3-2 8203. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem4 8192* Lemma for isfin3-2 8203. Being a chain, difference sets are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem5 8193* Lemma for isfin3-2 8203. There are infinite decrease points. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem6 8194* Lemma for isfin3-2 8203. Each K value is non-empty. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem7 8195* Lemma for isfin3-2 8203. Different K values are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem8 8196* Lemma for isfin3-2 8203. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.)

Theoremisf32lem9 8197* Lemma for isfin3-2 8203. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremisf32lem10 8198* Lemma for isfin3-2 . Write in terms of weak dominance. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
*

Theoremisf32lem11 8199* Lemma for isfin3-2 8203. Remove hypotheses from isf32lem10 8198. (Contributed by Stefan O'Rear, 17-May-2015.)
*

Theoremisf32lem12 8200* Lemma for isfin3-2 8203. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
*

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