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Theorem List for Metamath Proof Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlimuni 4601 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)

Theoremlimuni2 4602 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)

Theorem0ellim 4603 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)

Theoremlimelon 4604 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)

Theoremonn0 4605 The class of all ordinal numbers in not empty. (Contributed by NM, 17-Sep-1995.)

Theoremsuceq 4606 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremelsuci 4607 Membership in a successor. This one-way implication does not require that either or be sets. (Contributed by NM, 6-Jun-1994.)

Theoremelsucg 4608 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)

Theoremelsuc2g 4609 Variant of membership in a successor, requiring that rather than be a set. (Contributed by NM, 28-Oct-2003.)

Theoremelsuc 4610 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)

Theoremelsuc2 4611 Membership in a successor. (Contributed by NM, 15-Sep-2003.)

Theoremnfsuc 4612 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)

Theoremelelsuc 4613 Membership in a successor. (Contributed by NM, 20-Jun-1998.)

Theoremsucel 4614* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)

Theoremsuc0 4615 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)

Theoremsucprc 4616 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)

Theoremunisuc 4617 A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)

Theoremsssucid 4618 A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)

Theoremsucidg 4619 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)

Theoremsucid 4620 A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)

Theoremnsuceq0 4621 No successor is empty. (Contributed by NM, 3-Apr-1995.)

Theoremeqelsuc 4622 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)

Theoremiunsuc 4623* Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremsuctr 4624 The successor of a transtive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)

Theoremtrsuc 4625 A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremtrsucss 4626 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)

Theoremordsssuc 4627 A subset of an ordinal belongs to its successor. (Contributed by NM, 28-Nov-2003.)

Theoremonsssuc 4628 A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)

Theoremordsssuc2 4629 An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremonmindif 4630 When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)

Theoremordnbtwn 4631 There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.)

Theoremonnbtwn 4632 There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.)

Theoremsucssel 4633 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)

Theoremorddif 4634 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)

Theoremorduniss 4635 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)

Theoremordtri2or 4636 A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremordtri2or2 4637 A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)

Theoremordtri2or3 4638 A consequence of total ordering for ordinal classes. Similar to ordtri2or2 4637. (Contributed by David Moews, 1-May-2017.)

Theoremordelinel 4639 The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.)

Theoremordssun 4640 Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)

Theoremordequn 4641 The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)

Theoremordun 4642 The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)

Theoremordunisssuc 4643 A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)

Theoremsuc11 4644 The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)

Theoremonordi 4645 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)

Theoremontrci 4646 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)

Theoremonirri 4647 An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Theoremoneli 4648 A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)

Theoremonelssi 4649 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)

Theoremonssneli 4650 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)

Theoremonssnel2i 4651 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)

Theoremonelini 4652 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)

Theoremoneluni 4653 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)

Theoremonunisuci 4654 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)

Theoremonsseli 4655 Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)

Theoremonun2i 4656 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)

Theoremunizlim 4657 An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)

Theoremon0eqel 4658 An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)

Theoremsnsn0non 4659 The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 4808). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 4916. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

2.4  ZF Set Theory - add the Axiom of Union

2.4.1  Introduce the Axiom of Union

Axiomax-un 4660* Axiom of Union. An axiom of Zermelo-Fraenkel set theory. It states that a set exists that includes the union of a given set i.e. the collection of all members of the members of . The variant axun2 4662 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4663. A version using class notation is uniex 4664.

The union of a class df-uni 3976 should not be confused with the union of two classes df-un 3285. Their relationship is shown in unipr 3989. (Contributed by NM, 23-Dec-1993.)

Theoremzfun 4661* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Theoremaxun2 4662* A variant of the Axiom of Union ax-un 4660. For any set , there exists a set whose members are exactly the members of the members of i.e. the union of . Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Theoremuniex2 4663* The Axiom of Union using the standard abbreviation for union. Given any set , its union exists. (Contributed by NM, 4-Jun-2006.)

Theoremuniex 4664 The Axiom of Union in class notation. This says that if is a set i.e. (see isset 2920), then the union of is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)

Theoremuniexg 4665 The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent instead of to make the theorem more general and thus shorten some proofs; obviously the universal class constant is one possible substitution for class variable . (Contributed by NM, 25-Nov-1994.)

Theoremunex 4666 The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)

Theoremtpex 4667 A triple of classes exists. (Contributed by NM, 10-Apr-1994.)

Theoremunexb 4668 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)

Theoremunexg 4669 A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)

Theoremunisn2 4670 A version of unisn 3991 without the hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)

Theoremunisn3 4671* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)

Theoremsnnex 4672* The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)

Theoremdifex2 4673 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 4674 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 4675* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)

Theoremeusv1 4676* Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.)

Theoremeusvnf 4677* 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 4678* Two ways to say that is a set expression that does not depend on . (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2i 4679* Two ways to express single-valuedness of a class expression . (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2nf 4680* Two ways to express single-valuedness of a class expression . (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremeusv2 4681* 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 4682* 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 4683* Lemma for reusv2 4688. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv2lem2 4684* Lemma for reusv2 4688. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv2lem3 4685* Lemma for reusv2 4688. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv2lem4 4686* Lemma for reusv2 4688. (Contributed by NM, 13-Dec-2012.)

Theoremreusv2lem5 4687* Lemma for reusv2 4688. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Theoremreusv2 4688* 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 4689* Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)

Theoremreusv3 4690* Two ways to express single-valuedness of a class expression . See reusv1 4682 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)

Theoremeusv4 4691* Two ways to express single-valuedness of a class expression . (Contributed by NM, 27-Oct-2010.)

Theoremreusv5OLD 4692* 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 4693* Two ways to express single-valuedness of a class expression . The converse does not hold. Note that means is a singleton (uniintsn 4047). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremreusv7OLD 4694* Two ways to express single-valuedness of a class expression . Note that means is a singleton (uniintsn 4047). (Contributed by NM, 14-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremalxfr 4695* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 18-Feb-2007.)

Theoremralxfrd 4696* 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 4697* 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 4698* Transfer universal quantification from a variable to another variable contained in expression . (Contributed by Mario Carneiro, 20-Aug-2014.)

Theoremrexxfr2d 4699* 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 4700* 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.)

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