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Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)
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2.6.6  Scott's trick; collection principle; Hilbert's epsilon

Theoremscottex 7801* Scott's trick collects all sets that have a certain property and are of the 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 7802* Scott's trick collects all sets that have a certain property and are of the 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 7803* Theorem scheme version of scottex 7801. The collection of all of minimum rank such that is true, is a set. (Contributed by NM, 13-Oct-2003.)

Theoremscott0s 7804* Theorem scheme version of scott0 7802. 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 7805* Lemma for the Collection Principle cp 7807. (Contributed by NM, 17-Oct-2003.)

Theoremcplem2 7806* -Lemma for the Collection Principle cp 7807. (Contributed by NM, 17-Oct-2003.)

Theoremcp 7807* 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 7801 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 7808* A very strong generalization of the Axiom of Replacement (compare zfrep6 5960), derived from the Collection Principle cp 7807. 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 7809* A variant of the Boundedness Axiom bnd 7808 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.)

Theoremkardex 7810* The collection of all sets equinumerous to a set and having the 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 7811* 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 the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 8418). 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 7810 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from . (Contributed by NM, 18-Dec-2003.)

Theoremhtalem 7812* 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 7813* 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 the 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 1644 and weth 8367, using scottexs 7803 to establish the existence of .

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 7812. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

2.6.7  Cardinal numbers

Syntaxccrd 7814 Extend class definition to include the cardinal size function.

Syntaxcale 7815 Extend class definition to include the aleph function.

Syntaxccf 7816 Extend class definition to include the cofinality function.

Syntaxwacn 7817 The axiom of choice for limited-length sequences.
AC

Definitiondf-card 7818* 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 8413 for its value, cardval2 7870 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 8418. 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 7819 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 7939, alephsuc 7941, and alephlim 7940. 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 7820* Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 8119 for its value and a description. (Contributed by NM, 1-Apr-2004.)

Definitiondf-acn 7821* 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 7822* 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 7823 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 7824* 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 7825 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremennum 7826 Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremfinnum 7827 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremonenon 7828 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremtskwe 7829* A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremxpnum 7830 The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremcardval3 7831* 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 7832 Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremisnum3 7833 A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremoncardval 7834* The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 8413, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremoncardid 7835 Any ordinal number is equinumerous to its cardinal number. Unlike cardid 8414, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)

Theoremcardonle 7836 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 7837 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)

Theoremcardidm 7838 The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremoncard 7839* 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 7840 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 7841 A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)

Theoremcardnn 7842 The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremcardnueq0 7843 The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremcardne 7844 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 7845 If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7846 are meant to replace carden 8418 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)

Theoremcarden2b 7846 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7845 are meant to replace carden 8418 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Theoremcard1 7847* A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremcardsn 7848 A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremcarddomi2 7849 Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8421, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremsdomsdomcardi 7850 A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremcardlim 7851 An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremcardsdomelir 7852 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7853 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremcardsdomel 7853 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 7854* Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremiscard2 7855* 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 7856 Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8421, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremharcard 7857 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 7858* Lemma for cardprc 7859. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremcardprc 7859 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 8428 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7505 to construct (effectively) from , which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)

Theoremcarduni 7860* 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 7861* The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)

Theoremcardennn 7862 If is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)

Theoremcardsucinf 7863 The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.)

Theoremcardsucnn 7864 The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 7863. (Contributed by NM, 7-Nov-2008.)

Theoremcardom 7865 The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.)

Theoremcarden2 7866 Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 8418, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)

Theoremcardsdom2 7867 A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremdomtri2 7868 Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremnnsdomel 7869 Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremcardval2 7870* An alternate version of the value of the cardinal number of a set. Compare cardval 8413. This theorem could be used to give us a simpler definition of in place of df-card 7818. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)

Theoremisinffi 7871* An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7314 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremfidomtri 7872 Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoremfidomtri2 7873 Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 7-May-2015.)

Theoremharsdom 7874 The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.)
har

Theoremonsdom 7875* Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.)

Theoremharval2 7876* An alternative expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
har

Theoremcardmin2 7877* The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)

Theorempm54.43lem 7878* In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 7847), so that their means, in our notation, . Here we show that this is equivalent to so that we can use the latter more convenient notation in pm54.43 7879. (Contributed by NM, 4-Nov-2013.)

Theorempm54.43 7879 Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2.

Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 7847), so that their means, in our notation, which is the same as by pm54.43lem 7878. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.)

Theorem pm110.643 8049 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)

Theorempr2nelem 7880 Lemma for pr2ne 7881. (Contributed by FL, 17-Aug-2008.)

Theorempr2ne 7881 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)

Theoremprdom2 7882 An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)

Theoremen2eqpr 7883 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremdif1card 7884 The cardinality of a non-empty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.)

Theoremleweon 7885* Lexicographical order is a well-ordering of . Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7886, this order is not set-like, as the preimage of is the proper class . (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoremr0weon 7886* A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoreminfxpenlem 7887* Lemma for infxpen 7888. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
OrdIso

Theoreminfxpen 7888 Every infinite ordinal is equinumerous to its cross product. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation is a well-ordering of with the additional property that -initial segments of (where is a limit ordinal) are of cardinality at most . (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremxpomen 7889 The cross product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)

Theoreminfxpidm2 7890 The cross product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 8429. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpenc 7891* A canonical version of infxpen 7888, by a completely different approach (although it uses infxpen 7888 via xpomen 7889). Using Cantor's normal form, we can show that respects equinumerosity (oef1o 7647), so that all the steps of can be verified using bijections to do the ordinal commutations. (The assumption on can be satisfied using cnfcom3c 7655.) (Contributed by Mario Carneiro, 30-May-2015.)
CNF CNF                             CNF CNF

Theoreminfxpenc2lem1 7892* Lemma for infxpenc2 7895. (Contributed by Mario Carneiro, 30-May-2015.)

Theoreminfxpenc2lem2 7893* Lemma for infxpenc2 7895. (Contributed by Mario Carneiro, 30-May-2015.)
CNF CNF                             CNF CNF

Theoreminfxpenc2lem3 7894* Lemma for infxpenc2 7895. (Contributed by Mario Carneiro, 30-May-2015.)

Theoreminfxpenc2 7895* Existence form of infxpenc 7891. A "uniform" or "canonical" version of infxpen 7888, asserting the existence of a single function that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremiunmapdisj 7896* The union is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)

Theoremfseqenlem1 7897* Lemma for fseqen 7900. (Contributed by Mario Carneiro, 17-May-2015.)
seq𝜔

Theoremfseqenlem2 7898* Lemma for fseqen 7900. (Contributed by Mario Carneiro, 17-May-2015.)
seq𝜔

Theoremfseqdom 7899* One half of fseqen 7900. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremfseqen 7900* A set that is equinumerous to its cross product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.)

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