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Type | Label | Description |
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Statement | ||
Theorem | cfsmo 8701* | The map in cff1 8688 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.) |
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Theorem | cfcoflem 8702* | Lemma for cfcof 8704, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.) |
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Theorem | coftr 8703* |
If there is a cofinal map from ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cfcof 8704* |
If there is a cofinal map from ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cfidm 8705 | The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.) |
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Theorem | alephsing 8706 |
The cofinality of a limit aleph is the same as the cofinality of its
argument, so if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sornom 8707* |
The range of a single-step monotone function from ![]() |
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Syntax | cfin1a 8708 | Extend class notation to include the class of Ia-finite sets. |
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Syntax | cfin2 8709 | Extend class notation to include the class of II-finite sets. |
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Syntax | cfin4 8710 | Extend class notation to include the class of IV-finite sets. |
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Syntax | cfin3 8711 | Extend class notation to include the class of III-finite sets. |
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Syntax | cfin5 8712 | Extend class notation to include the class of V-finite sets. |
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Syntax | cfin6 8713 | Extend class notation to include the class of VI-finite sets. |
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Syntax | cfin7 8714 | Extend class notation to include the class of VII-finite sets. |
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Definition | df-fin1a 8715* | 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 7573 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.) |
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Definition | df-fin2 8716* | 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.) |
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Definition | df-fin4 8717* | 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.) |
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Definition | df-fin3 8718 | 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.) |
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Definition | df-fin5 8719 |
A set is V-finite iff it behaves finitely under ![]() |
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Definition | df-fin6 8720 |
A set is VI-finite iff it behaves finitely under ![]() |
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Definition | df-fin7 8721* | 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.) |
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Theorem | isfin1a 8722* | Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | fin1ai 8723 | Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | isfin2 8724* | Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | fin2i 8725 | Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | isfin3 8726 | Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | isfin4 8727* | Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | fin4i 8728 | Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | isfin5 8729 | Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | isfin6 8730 | Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | isfin7 8731* | Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
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Theorem | sdom2en01 8732 | A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | infpssrlem1 8733 | Lemma for infpssr 8738. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | infpssrlem2 8734 | Lemma for infpssr 8738. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | infpssrlem3 8735 | Lemma for infpssr 8738. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | infpssrlem4 8736 | Lemma for infpssr 8738. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | infpssrlem5 8737 | Lemma for infpssr 8738. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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Theorem | infpssr 8738 | 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.) |
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Theorem | fin4en1 8739 | Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
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Theorem | ssfin4 8740 | 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.) |
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Theorem | domfin4 8741 | A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.) |
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Theorem | ominf4 8742 |
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Theorem | infpssALT 8743* | Alternate proof of infpss 8647, shorter but requiring Replacement (ax-rep 4515). (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | isfin4-2 8744 | 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.) |
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Theorem | isfin4-3 8745 | Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 8727 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.) |
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Theorem | fin23lem7 8746* | Lemma for isfin2-2 8749. 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.) |
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Theorem | fin23lem11 8747* | Lemma for isfin2-2 8749. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
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Theorem | fin2i2 8748 | A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.) |
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Theorem | isfin2-2 8749* | FinII expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.) |
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Theorem | ssfin2 8750 | 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.) |
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Theorem | enfin2i 8751 | II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.) |
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Theorem | fin23lem24 8752 | Lemma for fin23 8819. 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.) |
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Theorem | fincssdom 8753 | In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
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Theorem | fin23lem25 8754 | Lemma for fin23 8819. In a chain of finite sets, equinumerosity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
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Theorem | fin23lem26 8755* | Lemma for fin23lem22 8757. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
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Theorem | fin23lem23 8756* | Lemma for fin23lem22 8757. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
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Theorem | fin23lem22 8757* |
Lemma for fin23 8819 but could be used elsewhere if we find a good
name for
it. Explicit construction of a bijection (actually an isomorphism, see
fin23lem27 8758) between an infinite subset of ![]() ![]() |
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Theorem | fin23lem27 8758* | The mapping constructed in fin23lem22 8757 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
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Theorem | isfin3ds 8759* | Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.) |
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Theorem | ssfin3ds 8760* | A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014.) |
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Theorem | fin23lem12 8761* |
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 |
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Theorem | fin23lem13 8762* |
Lemma for fin23 8819. Each step of ![]() |
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Theorem | fin23lem14 8763* |
Lemma for fin23 8819. ![]() |
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Theorem | fin23lem15 8764* |
Lemma for fin23 8819. ![]() |
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Theorem | fin23lem16 8765* |
Lemma for fin23 8819. ![]() ![]() ![]() ![]() |
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Theorem | fin23lem19 8766* |
Lemma for fin23 8819. The first set in ![]() |
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Theorem | fin23lem20 8767* |
Lemma for fin23 8819. ![]() |
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Theorem | fin23lem17 8768* |
Lemma for fin23 8819. By ? Fin3DS ? , ![]() ![]() |
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Theorem | fin23lem21 8769* |
Lemma for fin23 8819. ![]() ![]() ![]() |
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Theorem | fin23lem28 8770* | Lemma for fin23 8819. The residual is also one-to-one. This preserves the induction invariant. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
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Theorem | fin23lem29 8771* | Lemma for fin23 8819. The residual is built from the same elements as the previous sequence. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
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Theorem | fin23lem30 8772* | Lemma for fin23 8819. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
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Theorem | fin23lem31 8773* | Lemma for fin23 8819. 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.) |
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Theorem | fin23lem32 8774* | Lemma for fin23 8819. Wrap the previous construction into a function to hide the hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
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Theorem | fin23lem33 8775* | Lemma for fin23 8819. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
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Theorem | fin23lem34 8776* |
Lemma for fin23 8819. Establish induction invariants on ![]() ![]() ![]() ![]() |
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Theorem | fin23lem35 8777* |
Lemma for fin23 8819. Strict order property of ![]() |
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Theorem | fin23lem36 8778* |
Lemma for fin23 8819. Weak order property of ![]() |
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Theorem | fin23lem38 8779* | Lemma for fin23 8819. The contradictory chain has no minimum. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
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Theorem | fin23lem39 8780* |
Lemma for fin23 8819. Thus, we have that ![]() ![]() |
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Theorem | fin23lem40 8781* | Lemma for fin23 8819. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.) |
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Theorem | fin23lem41 8782* | Lemma for fin23 8819. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
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Theorem | isf32lem1 8783* | Lemma for isfin3-2 8797. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isf32lem2 8784* | Lemma for isfin3-2 8797. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isf32lem3 8785* | Lemma for isfin3-2 8797. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isf32lem4 8786* | Lemma for isfin3-2 8797. Being a chain, difference sets are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isf32lem5 8787* | Lemma for isfin3-2 8797. There are infinite decrease points. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isf32lem6 8788* | Lemma for isfin3-2 8797. Each K value is nonempty. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isf32lem7 8789* | Lemma for isfin3-2 8797. Different K values are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isf32lem8 8790* | Lemma for isfin3-2 8797. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.) |
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Theorem | isf32lem9 8791* | Lemma for isfin3-2 8797. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
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Theorem | isf32lem10 8792* | 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.) |
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Theorem | isf32lem11 8793* | Lemma for isfin3-2 8797. Remove hypotheses from isf32lem10 8792. (Contributed by Stefan O'Rear, 17-May-2015.) |
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Theorem | isf32lem12 8794* | Lemma for isfin3-2 8797. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
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Theorem | isfin32i 8795 | One half of isfin3-2 8797. (Contributed by Mario Carneiro, 3-Jun-2015.) |
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Theorem | isf33lem 8796* | Lemma for isfin3-3 8798. (Contributed by Stefan O'Rear, 17-May-2015.) |
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Theorem | isfin3-2 8797 |
Weakly Dedekind-infinite sets are exactly those which can be mapped onto
![]() |
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Theorem | isfin3-3 8798* |
Weakly Dedekind-infinite sets are exactly those with an ![]() |
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Theorem | fin33i 8799* |
Inference from isfin3-3 8798. (This is actually a bit stronger than
isfin3-3 8798 because it does not assume ![]() |
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Theorem | compsscnvlem 8800* | Lemma for compsscnv 8801. (Contributed by Mario Carneiro, 17-May-2015.) |
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