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Theorem List for Metamath Proof Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremomelon 7601 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)

Theoremdfom3 7602* The class of natural numbers omega can be defined as the smallest "inductive set," which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. (Contributed by NM, 6-Aug-1994.)

Theoremelom3 7603* A simplification of elom 4848 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)

Theoremdfom4 7604* A simplification of df-om 4846 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)

Theoremdfom5 7605 is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)

Theoremoancom 7606 Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)

Theoremisfinite 7607 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.)

Theoremnnsdom 7608 A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.)

Theoremomenps 7609 Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.)

Theoremomensuc 7610 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)

Theoreminfdifsn 7611 Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)

Theoreminfdiffi 7612 Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremunbnn3 7613* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 7363 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)

Theoremnoinfep 7614* Using the Axiom of Regularity in the form zfregfr 7570, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)

TheoremnoinfepOLD 7615* Using the Axiom of Regularity in the form zfregfr 7570, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

2.6.3  Cantor normal form

Syntaxccnf 7616 Extend class notation with the Cantor normal form function.
CNF

Definitiondf-cnf 7617* Define the Cantor normal form function, which takes as input a finitely supported function from to and outputs the corresponding member of the ordinal exponential . The content of the original Cantor Normal Form theorem is that for this function is a bijection onto for any ordinal (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to ). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 7651 of this function in terms of df-oi 7479. (Contributed by Mario Carneiro, 25-May-2015.)
CNF OrdIso seq𝜔

Theoremcantnffval 7618* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
CNF OrdIso seq𝜔

Theoremcantnfdm 7619* The domain of the Cantor normal form function (in later lemmas we will use CNF to abbreviate "the set of finitely supported functions from to "). (Contributed by Mario Carneiro, 25-May-2015.)
CNF

Theoremcantnfvalf 7620* Lemma for cantnf 7649. The function appearing in cantnfval 7623 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
seq𝜔

Theoremcantnfs 7621 Elementhood in the set of finitely supported functions from to . (Contributed by Mario Carneiro, 25-May-2015.)
CNF

Theoremcantnfcl 7622 Basic properties of the order isomorphism used later. The support of an is a finite subset of , so it is well-ordered by and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                      OrdIso

Theoremcantnfval 7623* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                      OrdIso               seq𝜔        CNF

Theoremcantnfval2 7624* Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                      OrdIso               seq𝜔        CNF seq𝜔

Theoremcantnfsuc 7625* The value of the recursive function at a successor. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                      OrdIso               seq𝜔

Theoremcantnfle 7626* A lower bound on the CNF function. Since CNF is defined as the sum of over all in the support of , it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all instead of just those in the support). (Contributed by Mario Carneiro, 28-May-2015.)
CNF                      OrdIso               seq𝜔               CNF

Theoremcantnflt 7627* An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent where is larger than any exponent which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                      OrdIso               seq𝜔

Theoremcantnflt2 7628 An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                                  CNF

Theoremcantnff 7629 The CNF function is a function from finitely supported functions from to , to the ordinal exponential . (Contributed by Mario Carneiro, 28-May-2015.)
CNF                      CNF

Theoremcantnf0 7630 The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
CNF                             CNF

Theoremcantnfreslem 7631* The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.)
CNF

Theoremcantnfrescl 7632* A function is finitely supported from to iff the extended function is finitely supported from to . (Contributed by Mario Carneiro, 25-May-2015.)
CNF                                                  CNF

Theoremcantnfres 7633* The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                                                  CNF               CNF CNF

Theoremcantnfp1lem1 7634* Lemma for cantnfp1 7637. (Contributed by Mario Carneiro, 20-Jun-2015.)
CNF

Theoremcantnfp1lem2 7635* Lemma for cantnfp1 7637. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                                                OrdIso

Theoremcantnfp1lem3 7636* Lemma for cantnfp1 7637. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                                                OrdIso        seq𝜔        OrdIso        seq𝜔        CNF CNF

Theoremcantnfp1 7637* If is created by adding a single term to , where is larger than any element of the support of , then is also a finitely supported function and it is assigned the value where is the value of . (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                                         CNF CNF

Theoremoemapso 7638* The relation is a strict order on (a corollary of wemapso2 7521). (Contributed by Mario Carneiro, 28-May-2015.)
CNF

Theoremoemapval 7639* Value of the relation . (Contributed by Mario Carneiro, 28-May-2015.)
CNF

Theoremoemapvali 7640* If , then there is some witnessing this, but we can say more and in fact there is a definable expression that also witnesses . (Contributed by Mario Carneiro, 25-May-2015.)
CNF

Theoremcantnflem1a 7641* Lemma for cantnf 7649. (Contributed by Mario Carneiro, 4-Jun-2015.)
CNF

Theoremcantnflem1b 7642* Lemma for cantnf 7649. (Contributed by Mario Carneiro, 4-Jun-2015.)
CNF                                                         OrdIso

Theoremcantnflem1c 7643* Lemma for cantnf 7649. (Contributed by Mario Carneiro, 4-Jun-2015.)
CNF                                                         OrdIso

Theoremcantnflem1d 7644* Lemma for cantnf 7649. (Contributed by Mario Carneiro, 4-Jun-2015.)
CNF                                                         OrdIso        seq𝜔        CNF

Theoremcantnflem1 7645* Lemma for cantnf 7649. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct are -related as or , and WLOG assuming that , we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                                         OrdIso        seq𝜔        CNF CNF

Theoremcantnflem2 7646* Lemma for cantnf 7649. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                    CNF

Theoremcantnflem3 7647* Lemma for cantnf 7649. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than has a normal form, we can use oeeu 6846 to factor into the form where and (and a fortiori ). Then since , has a normal form, and by appending the term using cantnfp1 7637 we get a normal form for . (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                    CNF                                                  CNF               CNF

Theoremcantnflem4 7648* Lemma for cantnf 7649. Complete the induction step of cantnflem3 7647. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                                    CNF                                           CNF

Theoremcantnf 7649* The Cantor Normal Form theorem. The function CNF , which maps a finitely supported function from to to the sum over all indexes such that is nonzero, is an order isomorphism from the ordering of finitely supported functions to the set under the natural order. Setting and letting be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 7633, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)
CNF                             CNF

Theoremoemapwe 7650* The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternative definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                             OrdIso

Theoremcantnffval2 7651* An alternative definition of df-cnf 7617 which relies on cantnf 7649. (Note that although the use of seems self-referential, one can use cantnfdm 7619 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
CNF                             CNF OrdIso

Theoremcantnff1o 7652 Simplify the isomorphism of cantnf 7649 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
CNF                      CNF

Theoremmapfien 7653* A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremwemapwe 7654* Construct lexicographic order on a function space based on a reverse well-ordering of the indexes and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.)
OrdIso        OrdIso

Theoremoef1o 7655* A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption can be discharged using fveqf1o 6029.) (Contributed by Mario Carneiro, 30-May-2015.)
CNF CNF

Theoremcnfcomlem 7656* Lemma for cnfcom 7657. (Contributed by Mario Carneiro, 30-May-2015.)
CNF                      CNF        OrdIso        seq𝜔        seq𝜔

Theoremcnfcom 7657* Any ordinal is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.)
CNF                      CNF        OrdIso        seq𝜔        seq𝜔

Theoremcnfcom2lem 7658* Lemma for cnfcom2 7659. (Contributed by Mario Carneiro, 30-May-2015.)
CNF                      CNF        OrdIso        seq𝜔        seq𝜔

Theoremcnfcom2 7659* Any nonzero ordinal is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.)
CNF                      CNF        OrdIso        seq𝜔        seq𝜔

Theoremcnfcom3lem 7660* Lemma for cnfcom3 7661. (Contributed by Mario Carneiro, 30-May-2015.)
CNF                      CNF        OrdIso        seq𝜔        seq𝜔

Theoremcnfcom3 7661* Any infinite ordinal is equinumerous to a power of . (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 7663.) (Contributed by Mario Carneiro, 28-May-2015.)
CNF                      CNF        OrdIso        seq𝜔        seq𝜔

Theoremcnfcom3clem 7662* Lemma for cnfcom3c 7663. (Contributed by Mario Carneiro, 30-May-2015.)
CNF        CNF        OrdIso        seq𝜔        seq𝜔

Theoremcnfcom3c 7663* Wrap the construction of cnfcom3 7661 into an existence quantifier. For any , there is a bijection from to some power of . Furthermore, this bijection is canonical , which means that we can find a single function which will give such bijections for every less than some arbitrarily large bound . (Contributed by Mario Carneiro, 30-May-2015.)

2.6.4  Transitive closure

Theoremtrcl 7664* For any set , show the properties of its transitive closure . Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 7665 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremtz9.1 7665* Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 7664 for an explicit expression for the transitive closure. Apparently open problems are whether this theorem can be proved without the Axiom of Infinity; if not, then whether it implies Infinity; and if not, what is the "property" that Infinity has that the other axioms don't have that is weaker than Infinity itself?

(Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.)

Theoremtz9.1c 7666* Alternative expression for the existence of transitive closures tz9.1 7665: the intersection of all transitive sets containing is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)

Theoremepfrs 7667* The strong form of the Axiom of Regularity (no sethood requirement on ), with the axiom itself present as an antecedent. See also zfregs 7668. (Contributed by Mario Carneiro, 22-Mar-2013.)

Theoremzfregs 7668* The strong form of the Axiom of Regularity, which does not require that be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 7667. (Contributed by NM, 17-Sep-2003.)

Theoremzfregs2 7669* Alternate strong form of the Axiom of Regularity. Not every element of a non-empty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)

Theoremen3lplem1 7670* Lemma for en3lp 7672. (Contributed by Alan Sare, 28-Oct-2011.)

Theoremen3lplem2 7671* Lemma for en3lp 7672. (Contributed by Alan Sare, 28-Oct-2011.)

Theoremen3lp 7672 No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 28957 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremsetind 7673* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)

Theoremsetind2 7674 Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)

Syntaxctc 7675 Extend class notation to include the transitive closure function.

Definitiondf-tc 7676* The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcvalg 7677* Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 7593; see tz9.1 7665.) (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcid 7678 Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtctr 7679 Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcmin 7680 Defining property of the transitive closure function: it is a subset of any transitive class containing . (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtc2 7681* A variant of the definition of the transitive closure function, using instead the smallest transitive set containing as a member, gives almost the same set, except that itself must be added because it is not usually a member of (and it is never a member if is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcsni 7682 The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)

Theoremtcss 7683 The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcel 7684 The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcidm 7685 The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtc0 7686 The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)

Theoremtc00 7687 The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)

2.6.5  Rank

Syntaxcr1 7688 Extend class definition to include the cumulative hierarchy of sets function.

Syntaxcrnk 7689 Extend class definition to include rank function.

Definitiondf-r1 7690 Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation (). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 7717). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 7694, r1suc 7696, and r1lim 7698. Theorem r1val1 7712 shows a recursive definition that works for all values, and theorems r1val2 7763 and r1val3 7764 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477), with a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.)

Definitiondf-rank 7691* Define the rank function. See rankval 7742, rankval2 7744, rankval3 7766, or rankval4 7793 its value. The rank is a kind of "inverse" of the cumulative hierarchy of sets function : given a set, it returns an ordinal number telling us the smallest layer of the hierarchy to which the set belongs. Based on Definition 9.14 of [TakeutiZaring] p. 79. Theorem rankid 7759 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 7762. (Contributed by NM, 11-Oct-2003.)

Theoremr1funlim 7692 The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 7693 avoids ax-rep 4320.) (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1fnon 7693 The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremr10 7694 Value of the cumulative hierarchy of sets function at . Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremr1sucg 7695 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1suc 7696 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremr1limg 7697* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1lim 7698* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremr1fin 7699 The first levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.)

Theoremr1sdom 7700 Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)

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