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Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-fin7 7801* 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 7802* Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIa

Theoremfin1ai 7803 Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIa

Theoremisfin2 7804* Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinII []

Theoremfin2i 7805 Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinII []

Theoremisfin3 7806 Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIII FinIV

Theoremisfin4 7807* Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIV

Theoremfin4i 7808 Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
FinIV

Theoremisfin5 7809 Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinV

Theoremisfin6 7810 Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinVI

Theoremisfin7 7811* Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
FinVII

Theoremsdom2en01 7812 A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem1 7813 Lemma for infpssr 7818. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem2 7814 Lemma for infpssr 7818. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem3 7815 Lemma for infpssr 7818. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem4 7816 Lemma for infpssr 7818. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssrlem5 7817 Lemma for infpssr 7818. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoreminfpssr 7818 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 7819 Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
FinIV FinIV

Theoremssfin4 7820 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 7821 A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)
FinIV FinIV

Theoremominf4 7822 is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
FinIV

TheoreminfpssALT 7823* A set with a denumerable subset has a proper subset equinumerous to it, proved without AC or Infinity. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.)

Theoremisfin4-2 7824 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 7825 Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus the proper subset referred to in isfin4 7807 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
FinIV

Theoremfin23lem7 7826* Lemma for isfin2-2 7829. 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 7827* Lemma for isfin2-2 7829. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Theoremfin2i2 7828 A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)
FinII []

Theoremisfin2-2 7829* 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 7830 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 7831 II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
FinII FinII

Theoremfin23lem24 7832 Lemma for fin23 7899. 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 7833 In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.)

Theoremfin23lem25 7834 Lemma for fin23 7899. In a chain of finite sets, equinumerousity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfin23lem26 7835* Lemma for fin23lem22 7837. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfin23lem23 7836* Lemma for fin23lem22 7837. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfin23lem22 7837* Lemma for fin23 7899 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 7838) between an infinite subset of and itself. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremfin23lem27 7838* The mapping constructed in fin23lem22 7837 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremisfin3ds 7839* Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.)

Theoremssfin3ds 7840* A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014.)

Theoremfin23lem12 7841* 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 7842* Lemma for fin23 7899. Each step of is a decrease. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem14 7843* Lemma for fin23 7899. will never evolve to an empty set if it did not start with one. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem15 7844* Lemma for fin23 7899. is a monotone function. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem16 7845* Lemma for fin23 7899. 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 7846* Lemma for fin23 7899. 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 7847* Lemma for fin23 7899. is either contained in or disjoint from all input sets. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremfin23lem17 7848* Lemma for fin23 7899. 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 7849* Lemma for fin23 7899. 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 7850* Lemma for fin23 7899. The residual is also one-to-one. This preserves the induction invariant. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem29 7851* Lemma for fin23 7899. The residual is built from the same elements as the previous sequence. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem30 7852* Lemma for fin23 7899. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem31 7853* Lemma for fin23 7899. 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 7854* Lemma for fin23 7899. Wrap the previous construction into a function to hide the hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)
seq𝜔

Theoremfin23lem33 7855* Lemma for fin23 7899. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremfin23lem34 7856* Lemma for fin23 7899. 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 7857* Lemma for fin23 7899. Strict order property of . (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremfin23lem36 7858* Lemma for fin23 7899. Weak order property of . (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremfin23lem38 7859* Lemma for fin23 7899. The contradictory chain has no minimum. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Theoremfin23lem39 7860* Lemma for fin23 7899. Thus we have that could not have been in after all. (Contributed by Stefan O'Rear, 4-Nov-2014.)

Theoremfin23lem40 7861* Lemma for fin23 7899. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)
FinII

Theoremfin23lem41 7862* Lemma for fin23 7899. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.)
FinIII

Theoremisf32lem1 7863* Lemma for isfin3-2 7877. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem2 7864* Lemma for isfin3-2 7877. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem3 7865* Lemma for isfin3-2 7877. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem4 7866* Lemma for isfin3-2 7877. Being a chain, difference sets are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem5 7867* Lemma for isfin3-2 7877. There are infinite decrease points. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem6 7868* Lemma for isfin3-2 7877. Each K value is non-empty. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem7 7869* Lemma for isfin3-2 7877. Different K values are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisf32lem8 7870* Lemma for isfin3-2 7877. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.)

Theoremisf32lem9 7871* Lemma for isfin3-2 7877. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremisf32lem10 7872* 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 7873* Lemma for isfin3-2 7877. Remove hypotheses from isf32lem10 7872. (Contributed by Stefan O'Rear, 17-May-2015.)
*

Theoremisf32lem12 7874* Lemma for isfin3-2 7877. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
*

Theoremisfin32i 7875 One half of isfin3-2 7877. (Contributed by Mario Carneiro, 3-Jun-2015.)
FinIII *

Theoremisf33lem 7876* Lemma for isfin3-3 7878. (Contributed by Stefan O'Rear, 17-May-2015.)
FinIII

Theoremisfin3-2 7877 Weakly Dedekind-infinite sets are exactly those which can be mapped onto . (Contributed by Stefan O'Rear, 6-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
FinIII *

Theoremisfin3-3 7878* Weakly Dedekind-infinite sets are exactly those with an -indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.)
FinIII

Theoremfin33i 7879* Inference from isfin3-3 7878. (This is actually a bit stronger than isfin3-3 7878 because it does not assume is a set and does not use the Axiom of Infinity either.) (Contributed by Mario Carneiro, 17-May-2015.)
FinIII

Theoremcompsscnvlem 7880* Lemma for compsscnv 7881. (Contributed by Mario Carneiro, 17-May-2015.)

Theoremcompsscnv 7881* Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)

Theoremisf34lem1 7882* Lemma for isfin3-4 7892. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Theoremisf34lem2 7883* Lemma for isfin3-4 7892. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Theoremcompssiso 7884* Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
[] []

Theoremisf34lem3 7885* Lemma for isfin3-4 7892. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Theoremcompss 7886* Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

Theoremisf34lem4 7887* Lemma for isfin3-4 7892. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Theoremisf34lem5 7888* Lemma for isfin3-4 7892. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Theoremisf34lem7 7889* Lemma for isfin3-4 7892. (Contributed by Stefan O'Rear, 7-Nov-2014.)
FinIII

Theoremisf34lem6 7890* Lemma for isfin3-4 7892. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
FinIII

Theoremfin34i 7891* Inference from isfin3-4 7892. (Contributed by Mario Carneiro, 17-May-2015.)
FinIII

Theoremisfin3-4 7892* Weakly Dedekind-infinite sets are exactly those with an -indexed ascending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
FinIII

Theoremfin11a 7893 Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
FinIa

Theoremenfin1ai 7894 Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
FinIa FinIa

Theoremisfin1-2 7895 A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
FinIV

Theoremisfin1-3 7896 A set is I-finite iff every system of subsets contains a maximal subset. Definition I of [Levy58] p. 2. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
[]

Theoremisfin1-4 7897 A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
[]

Theoremdffin1-5 7898 Compact quantifier-free version of the standard definition df-fin 6753. (Contributed by Stefan O'Rear, 6-Jan-2015.)

Theoremfin23 7899 Every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets). The proof here is the only one I could find, from http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf p.94 (writeup by Tarski, credited to Kuratowski). Translated into English and modern notation, the proof proceeds as follows (variables renamed for uniqueness):

Suppose for a contradiction that is a set which is II-finite but not III-finite.

For any countable sequence of distinct subsets of , we can form a decreasing sequence of non-empty subsets by taking finite intersections of initial segments of while skipping over any element of which would cause the intersection to be empty.

By II-finiteness (as fin2i2 7828) this sequence contains its intersection, call it ; since by induction every subset in the sequence is non-empty, the intersection must be non-empty.

Suppose that an element of has non-empty intersection with . Thus said element has a non-empty intersection with the corresponding element of , therefore it was used in the construction of and all further elements of are subsets of , thus contains the . That is, all elements of either contain or are disjoint from it.

Since there are only two cases, there must exist an infinite subset of which uniformly either contain or are disjoint from it. In the former case we can create an infinite set by subtracting from each element. In either case, call the result ; this is an infinite set of subsets of , each of which is disjoint from and contained in the union of ; the union of is strictly contained in the union of , because only the latter is a superset of the non-empty set .

The preceeding four steps may be iterated a countable number of times starting from the assumed denumerable set of subsets to produce a denumerable sequence of the sets from each stage. Great caution is required to avoid ax-dc 7956 here; in particular an effective version of the pigeonhole principle (for aleph-null pigeons and 2 holes) is required. Since a denumerable set of subsets is assumed to exist, we can conclude without the axiom.

This sequence is strictly decreasing, thus it has no minimum, contradicting the first assumption. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

FinII FinIII

Theoremfin34 7900 Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
FinIII FinIV

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