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Theorem List for Metamath Proof Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremunexb 4701 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)

Theoremunexg 4702 A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)

Theoremunisn2 4703 A version of unisn 4023 without the hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)

Theoremunisn3 4704* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)

Theoremsnnex 4705* The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)

Theoremdifex2 4706 If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremopeluu 4707 Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)

Theoremuniuni 4708* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)

Theoremeusv1 4709* Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.)

Theoremeusvnf 4710* Even if is free in , it is effectively bound when is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremeusvnfb 4711* Two ways to say that is a set expression that does not depend on . (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2i 4712* Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2nf 4713* Two ways to express single-valuedness of a class expression . (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2 4714* Two ways to express single-valuedness of a class expression . (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremreusv1 4715* Two ways to express single-valuedness of a class expression . (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremreusv2lem1 4716* Lemma for reusv2 4721. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv2lem2 4717* Lemma for reusv2 4721. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv2lem3 4718* Lemma for reusv2 4721. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv2lem4 4719* Lemma for reusv2 4721. (Contributed by NM, 13-Dec-2012.)

Theoremreusv2lem5 4720* Lemma for reusv2 4721. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv2 4721* Two ways to express single-valuedness of a class expression that is constant for those such that . The first antecedent ensures that the constant value belongs to the existential uniqueness domain , and the second ensures that is evaluated for at least one . (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv3i 4722* Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)

Theoremreusv3 4723* Two ways to express single-valuedness of a class expression . See reusv1 4715 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)

Theoremeusv4 4724* Two ways to express single-valuedness of a class expression . (Contributed by NM, 27-Oct-2010.)

Theoremreusv5OLD 4725* Two ways to express single-valuedness of a class expression . (Contributed by NM, 16-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremreusv6OLD 4726* Two ways to express single-valuedness of a class expression . The converse does not hold. Note that means is a singleton (uniintsn 4079). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremreusv7OLD 4727* Two ways to express single-valuedness of a class expression . Note that means is a singleton (uniintsn 4079). (Contributed by NM, 14-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremalxfr 4728* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 18-Feb-2007.)

Theoremralxfrd 4729* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremrexxfrd 4730* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)

Theoremralxfr2d 4731* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.)

Theoremrexxfr2d 4732* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremralxfr 4733* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

TheoremralxfrALT 4734* Transfer universal quantification from a variable to another variable contained in expression . This proof does not use ralxfrd 4729. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremrexxfr 4735* Transfer existence from a variable to another variable contained in expression . (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

Theoremrabxfrd 4736* Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 16-Jan-2012.)

Theoremrabxfr 4737* Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 10-Jun-2005.)

Theoremreuxfr2d 4738* Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 16-Jan-2012.) (Revised by NM, 16-Jun-2017.)

Theoremreuxfr2 4739* Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)

Theoremreuxfrd 4740* Transfer existential uniqueness from a variable to another variable contained in expression . Use reuhypd 4742 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)

Theoremreuxfr 4741* Transfer existential uniqueness from a variable to another variable contained in expression . Use reuhyp 4743 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)

Theoremreuhypd 4742* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6573. (Contributed by NM, 16-Jan-2012.)

Theoremreuhyp 4743* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4741. (Contributed by NM, 15-Nov-2004.)

Theoremuniexb 4744 The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)

Theorempwexb 4745 The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)

Theoremuniv 4746 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Theoremeldifpw 4747 Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)

Theoremelpwun 4748 Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)

Theoremelpwunsn 4749 Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)

Theoremop1stb 4750 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5345 to extract the second member, op1sta 5343 for an alternate version, and op1st 6347 for the preferred version.) (Contributed by NM, 25-Nov-2003.)

Theoremiunpw 4751* An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)

Theoremfr3nr 4752 A well-founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 10-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Theoremepne3 4753 A set well-founded by epsilon contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Theoremdfwe2 4754* Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

2.4.2  Ordinals (continued)

Theoremordon 4755 The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)

Theoremepweon 4756 The epsilon relation well-orders the class of ordinal numbers. Proposition 4.8(g) of [Mendelson] p. 244. (Contributed by NM, 1-Nov-2003.)

Theoremonprc 4757 No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4755), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)

Theoremssorduni 4758 The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremssonuni 4759 The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)

Theoremssonunii 4760 The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)

Theoremordeleqon 4761 A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse. (Contributed by NM, 1-Jun-2003.)

Theoremordsson 4762 Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremonss 4763 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)

Theoremssonprc 4764 Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)

Theoremonuni 4765 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)

Theoremorduni 4766 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)

Theoremonint 4767 The intersection (infimum) of a non-empty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)

Theoremonint0 4768 The intersection of a class of ordinal numbers is zero iff the class contains zero. (Contributed by NM, 24-Apr-2004.)

Theoremonssmin 4769* A non-empty class of ordinal numbers has the smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.)

Theoremonminesb 4770 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 29-Sep-2003.)

Theoremonminsb 4771 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)

Theoremoninton 4772 The intersection of a non-empty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by NM, 29-Jan-1997.)

Theoremonintrab 4773 The intersection of a class of ordinal numbers exists iff it is an ordinal number. (Contributed by NM, 6-Nov-2003.)

Theoremonintrab2 4774 An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.)

Theoremonnmin 4775 No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)

Theoremonnminsb 4776* An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. is the wff resulting from the substitution of for in wff . (Contributed by NM, 9-Nov-2003.)

Theoremoneqmin 4777* A way to show that an ordinal number equals the minimum of a non-empty collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)

Theorembm2.5ii 4778* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)

Theoremonminex 4779* If a wff is true for an ordinal number, there is the smallest ordinal number for which it is true. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Mario Carneiro, 20-Nov-2016.)

Theoremsucon 4780 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)

Theoremsucexb 4781 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)

Theoremsucexg 4782 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)

Theoremsucex 4783 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)

Theoremonmindif2 4784 The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed. (Contributed by NM, 20-Nov-2003.)

Theoremsuceloni 4785 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)

Theoremordsuc 4786 The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.)

Theoremordpwsuc 4787 The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005.)

Theoremonpwsuc 4788 The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.)

Theoremsucelon 4789 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)

Theoremordsucss 4790 The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)

Theoremonpsssuc 4791 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Theoremordelsuc 4792 A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)

Theoremonsucmin 4793* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)

Theoremordsucelsuc 4794 Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (Contributed by NM, 22-Jun-1998.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremordsucsssuc 4795 The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)

Theoremordsucuniel 4796 Given an element of the union of an ordinal , is an element of itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)

Theoremordsucun 4797 The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.)

Theoremordunpr 4798 The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)

Theoremordunel 4799 The maximum of two ordinals belongs to a third if each of them do. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremonsucuni 4800 A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.)

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