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Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtcrank 7701 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 7702* 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 7703* 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 7704* Theorem scheme version of scottex 7702. The collection of all of minimum rank such that is true, is a set. (Contributed by NM, 13-Oct-2003.)

Theoremscott0s 7705* Theorem scheme version of scott0 7703. 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 7706* Lemma for the Collection Principle cp 7708. (Contributed by NM, 17-Oct-2003.)

Theoremcplem2 7707* -Lemma for the Collection Principle cp 7708. (Contributed by NM, 17-Oct-2003.)

Theoremcp 7708* 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 7702 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 7709* A very strong generalization of the Axiom of Replacement (compare zfrep6 5868), derived from the Collection Principle cp 7708. 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 7710* A variant of the Boundedness Axiom bnd 7709 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.)

Theoremkardex 7711* 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 7712* 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 8320). 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 7711 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 7713* 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 7714* 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 1639 and weth 8269, using scottexs 7704 to establish the existence of .

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 7713. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

2.6.7  Cardinal numbers

Syntaxccrd 7715 Extend class definition to include the cardinal size function.

Syntaxcale 7716 Extend class definition to include the aleph function.

Syntaxccf 7717 Extend class definition to include the cofinality function.

Syntaxwacn 7718 The axiom of choice for limited-length sequences.
AC

Definitiondf-card 7719* 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 8315 for its value, cardval2 7771 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 8320. 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 7720 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 7840, alephsuc 7842, and alephlim 7841. 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 7721* Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 8020 for its value and a description. (Contributed by NM, 1-Apr-2004.)

Definitiondf-acn 7722* 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 7723* 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 7724 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 7725* 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 7726 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremennum 7727 Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremfinnum 7728 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremonenon 7729 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremtskwe 7730* A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremxpnum 7731 The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremcardval3 7732* 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 7733 Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremisnum3 7734 A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremoncardval 7735* The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 8315, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremoncardid 7736 Any ordinal number is equinumerous to its cardinal number. Unlike cardid 8316, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)

Theoremcardonle 7737 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 7738 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)

Theoremcardidm 7739 The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremoncard 7740* 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 7741 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 7742 A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)

Theoremcardnn 7743 The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremcardnueq0 7744 The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)

Theoremcardne 7745 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 7746 If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7747 are meant to replace carden 8320 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)

Theoremcarden2b 7747 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7746 are meant to replace carden 8320 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Theoremcard1 7748* A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremcardsn 7749 A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremcarddomi2 7750 Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8323, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremsdomsdomcardi 7751 A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremcardlim 7752 An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremcardsdomelir 7753 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7754 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremcardsdomel 7754 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 7755* Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremiscard2 7756* 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 7757 Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8323, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremharcard 7758 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 7759* Lemma for cardprc 7760. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremcardprc 7760 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 8330 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7406 to construct (effectively) from , which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)

Theoremcarduni 7761* 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 7762* The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)

Theoremcardennn 7763 If is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)

Theoremcardsucinf 7764 The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.)

Theoremcardsucnn 7765 The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 7764. (Contributed by NM, 7-Nov-2008.)

Theoremcardom 7766 The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.)

Theoremcarden2 7767 Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 8320, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)

Theoremcardsdom2 7768 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 7769 Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremnnsdomel 7770 Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremcardval2 7771* An alternate version of the value of the cardinal number of a set. Compare cardval 8315. This theorem could be used to give us a simpler definition of in place of df-card 7719. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)

Theoremisinffi 7772* An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7219 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremfidomtri 7773 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 7774 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 7775 The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.)
har

Theoremonsdom 7776* 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 7777* An alternative expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
har

Theoremcardmin2 7778* The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)

Theorempm54.43lem 7779* 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 7748), 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 7780. (Contributed by NM, 4-Nov-2013.)

Theorempm54.43 7780 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 7748), so that their means, in our notation, which is the same as by pm54.43lem 7779. 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 7950 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)

Theorempr2nelem 7781 Lemma for pr2ne 7782. (Contributed by FL, 17-Aug-2008.)

Theorempr2ne 7782 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)

Theoremprdom2 7783 An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)

Theoremen2eqpr 7784 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremdif1card 7785 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 7786* Lexicographical order is a well-ordering of . Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7787, this order is not set-like, as the preimage of is the proper class . (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoremr0weon 7787* 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 7788* Lemma for infxpen 7789. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
OrdIso

Theoreminfxpen 7789 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 7790 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 7791 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 8331. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpenc 7792* A canonical version of infxpen 7789, by a completely different approach (although it uses infxpen 7789 via xpomen 7790). Using Cantor's normal form, we can show that respects equinumerosity (oef1o 7548), so that all the steps of can be verified using bijections to do the ordinal commutations. (The assumption on can be satisfied using cnfcom3c 7556.) (Contributed by Mario Carneiro, 30-May-2015.)
CNF CNF                             CNF CNF

Theoreminfxpenc2lem1 7793* Lemma for infxpenc2 7796. (Contributed by Mario Carneiro, 30-May-2015.)

Theoreminfxpenc2lem2 7794* Lemma for infxpenc2 7796. (Contributed by Mario Carneiro, 30-May-2015.)
CNF CNF                             CNF CNF

Theoreminfxpenc2lem3 7795* Lemma for infxpenc2 7796. (Contributed by Mario Carneiro, 30-May-2015.)

Theoreminfxpenc2 7796* Existence form of infxpenc 7792. A "uniform" or "canonical" version of infxpen 7789, 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 7797* The union is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)

Theoremfseqenlem1 7798* Lemma for fseqen 7801. (Contributed by Mario Carneiro, 17-May-2015.)
seq𝜔

Theoremfseqenlem2 7799* Lemma for fseqen 7801. (Contributed by Mario Carneiro, 17-May-2015.)
seq𝜔

Theoremfseqdom 7800* One half of fseqen 7801. (Contributed by Mario Carneiro, 18-Nov-2014.)

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