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Theorem List for Metamath Proof Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremr1pw 7401 A stronger property of than rankpw 7399. The latter merely proves that of the successor is a power set, but here we prove that if is in the cumulative hierarchy, then is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremr1pwOLD 7402 A stronger property of than rankpw 7399. The latter merely proves that of the successor is a power set, but here we prove that if is in the cumulative hierarchy, then is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremr1pwcl 7403 The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)

Theoremrankssb 7404 The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankss 7405 The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankunb 7406 The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankprb 7407 The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremrankopb 7408 The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremrankuni2b 7409* The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 8-Jun-2013.)

Theoremranksn 7410 The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankuni2 7411* The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankun 7412 The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by NM, 26-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankpr 7413 The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankop 7414 The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 13-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremr1rankid 7415 Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankeq0b 7416 A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremrankeq0 7417 A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1id 7418 The rank of the hierarchy of an ordinal number is itself. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankuni 7419 The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1b 7420 A relationship between rank and . See rankr1a 7392 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremranksuc 7421 The rank of a successor. (Contributed by NM, 18-Sep-2006.)

Theoremrankuniss 7422 Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 30-Nov-2003.)

Theoremrankval4 7423* The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.)

Theoremrankbnd 7424* The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.)

Theoremrankbnd2 7425* The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.)

Theoremrankc1 7426* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)

Theoremrankc2 7427* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)

Theoremrankelun 7428 Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)

Theoremrankelpr 7429 Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankelop 7430 Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.)

Theoremrankxpl 7431 A lower bound on the rank of a cross product. (Contributed by NM, 18-Sep-2006.)

Theoremrankxpu 7432 An upper bound on the rank of a cross product. (Contributed by NM, 18-Sep-2006.)

Theoremrankxplim 7433 The rank of a cross product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 7436 for the successor case. (Contributed by NM, 19-Sep-2006.)

Theoremrankxplim2 7434 If the rank of a cross product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.)

Theoremrankxplim3 7435 The rank of a cross product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.)

Theoremrankxpsuc 7436 The rank of a cross product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 7433 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)

Theoremtcwf 7437 The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcrank 7438 This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below . Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below , constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of has a rank below the rank of , since intuitively it contains only the members of and the members of those and so on, but nothing "bigger" than . (Contributed by Mario Carneiro, 23-Jun-2013.)

2.6.6  Scott's trick; collection principle; Hilbert's epsilon

Theoremscottex 7439* Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.)

Theoremscott0 7440* Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. is empty). (Contributed by NM, 15-Oct-2003.)

Theoremscottexs 7441* Theorem scheme version of scottex 7439. The collection of all of minimum rank such that is true, is a set. (Contributed by NM, 13-Oct-2003.)

Theoremscott0s 7442* Theorem scheme version of scott0 7440. The collection of all of minimum rank such that is true, is not empty iff there is an such that holds. (Contributed by NM, 13-Oct-2003.)

Theoremcplem1 7443* Lemma for the Collection Principle cp 7445. (Contributed by NM, 17-Oct-2003.)

Theoremcplem2 7444* -Lemma for the Collection Principle cp 7445. (Contributed by NM, 17-Oct-2003.)

Theoremcp 7445* Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 7439 that collapses a proper class into a set of minimum rank. The wff can be thought of as . Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)

Theorembnd 7446* A very strong generalization of the Axiom of Replacement (compare zfrep6 5600), derived from the Collection Principle cp 7445. Its strength lies in the rather profound fact that does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.)

Theorembnd2 7447* A variant of the Boundedness Axiom bnd 7446 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.)

Theoremkardex 7448* The collection of all sets equinumerous to a set and having least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.)

Theoremkarden 7449* If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 8055). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 7448 justify the definition of kard. The restriction to least rank prevents the proper class that would result from . (Contributed by NM, 18-Dec-2003.)

Theoremhtalem 7450* Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom," described on that page, with the additional antecedent. The element is the epsilon that the theorem emulates. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremhta 7451* A ZFC emulation of Hilbert's transfinite axiom. The set has the properties of Hilbert's epsilon, except that it also depends on a well-ordering . This theorem arose from discussions with Raph Levien on 5-Mar-2004 about translating the HOL proof language, which uses Hilbert's epsilon. See http://us.metamath.org/downloads/choice.txt (copy of obsolete link http://ghilbert.org/choice.txt) and http://us.metamath.org/downloads/megillaward2005he.pdf.

Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires as an antecedent. Class collects the sets of least rank for which is true. Class , which emulates the epsilon, is the minimum element in a well-ordering on .

If a well-ordering on can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace with a dummy set variable, say , and attach as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, (which will have as a free variable) will no longer be present, and we can eliminate by applying exlimiv 2023 and weth 8006, using scottexs 7441 to establish the existence of .

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 7450. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

2.6.7  Cardinal numbers

Syntaxccrd 7452 Extend class definition to include the cardinal size function.

Syntaxcale 7453 Extend class definition to include the aleph function.

Syntaxccf 7454 Extend class definition to include the cofinality function.

Syntaxwacn 7455 The axiom of choice for limited-length sequences.
AC

Definitiondf-card 7456* Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 8052 for its value, cardval2 7508 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 8055. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)

Definitiondf-aleph 7457 Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as theorems aleph0 7577, alephsuc 7579, and alephlim 7578. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it. (Contributed by NM, 21-Oct-2003.)
har

Definitiondf-cf 7458* Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 7757 for its value and a description. (Contributed by NM, 1-Apr-2004.)

Definitiondf-acn 7459* Define a local and length-limited version of the axiom of choice. The definition of the predicate AC is that for all families of nonempty subsets of indexed on (i.e. functions ), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC

Theoremcardf2 7460* The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 20-Sep-2014.)

Theoremcardon 7461 The cardinal number of a set is an ordinal number. Proposition 10.6(1) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremisnum2 7462* A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoremisnumi 7463 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremennum 7464 Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremfinnum 7465 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremonenon 7466 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremtskwe 7467* A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremxpnum 7468 The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremcardval3 7469* An alternative definition of the value of that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoremcardid2 7470 Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremisnum3 7471 A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremoncardval 7472* The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 8052, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremoncardid 7473 Any ordinal number is equinumerous to its cardinal number. Unlike cardid 8053, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)

Theoremcardonle 7474 The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)

Theoremcard0 7475 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)

Theoremcardidm 7476 The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremoncard 7477* A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremficardom 7478 The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.)

Theoremficardid 7479 A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)

Theoremcardnn 7480 The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremcardnueq0 7481 The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremcardne 7482 No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)

Theoremcarden2a 7483 If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7484 are meant to replace carden 8055 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)

Theoremcarden2b 7484 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7483 are meant to replace carden 8055 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Theoremcard1 7485* A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremcardsn 7486 A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremcarddomi2 7487 Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8058, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremsdomsdomcardi 7488 A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremcardlim 7489 An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremcardsdomelir 7490 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7491 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremcardsdomel 7491 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)

Theoremiscard 7492* Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremiscard2 7493* Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremcarddom2 7494 Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8058, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremharcard 7495 The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
har har

Theoremcardprclem 7496* Lemma for cardprc 7497. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremcardprc 7497 The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 8065 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7143 to construct (effectively) from , which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)

Theoremcarduni 7498* The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.)

Theoremcardiun 7499* The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)

Theoremcardennn 7500 If is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)

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