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Theorem List for Metamath Proof Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-pw 3801* Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of . When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if , then (ex-pw 21737). We will later introduce the Axiom of Power Sets ax-pow 4377, which can be expressed in class notation per pwexg 4383. Still later we will prove, in hashpw 11699, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.)

Theorempweq 3802 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)

Theorempweqi 3803 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)

Theorempweqd 3804 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)

Theoremelpw 3805 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)

Theoremelpwg 3806 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4363. (Contributed by NM, 6-Aug-2000.)

Theoremelpwi 3807 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)

Theoremelpwid 3808 An element of a power class is a subclass. Deduction form of elpwi 3807. (Contributed by David Moews, 1-May-2017.)

Theoremelelpwi 3809 If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.)

Theoremnfpw 3810 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theorempwidg 3811 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theorempwid 3812 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)

Theorempwss 3813* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)

2.1.17  Unordered and ordered pairs

Syntaxcsn 3814 Extend class notation to include singleton.

Syntaxcpr 3815 Extend class notation to include unordered pair.

Syntaxctp 3816 Extend class notation to include unordered triplet.

Syntaxcop 3817 Extend class notation to include ordered pair.

Syntaxcotp 3818 Extend class notation to include ordered triple.

Theoremsnjust 3819* Soundness justification theorem for df-sn 3820. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Definitiondf-sn 3820* Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of , although it is not very meaningful in this case. For an alternate definition see dfsn2 3828. (Contributed by NM, 5-Aug-1993.)

Definitiondf-pr 3821 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, (ex-pr 21738). They are unordered, so as proven by prcom 3882. For a more traditional definition, but requiring a dummy variable, see dfpr2 3830. (Contributed by NM, 5-Aug-1993.)

Definitiondf-tp 3822 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)

Definitiondf-op 3823* Definition of an ordered pair, equivalent to Kuratowski's definition when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 4006, opprc2 4007, and 0nelop 4446). For Kuratowski's actual definition when the arguments are sets, see dfop 3983. For the justifying theorem (for sets) see opth 4435. See dfopif 3981 for an equivalent formulation using the operation.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3823 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3823 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses.

There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition _2 , justified by opthwiener 4458. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition _3 is justified by opthreg 7573, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is _4 , justified by opthprc 4925. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 11563. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 9992. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

Definitiondf-ot 3824 Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)

Theoremsneq 3825 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)

Theoremsneqi 3826 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)

Theoremsneqd 3827 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)

Theoremdfsn2 3828 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Theoremelsn 3829* There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)

Theoremdfpr2 3830* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Theoremelprg 3831 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)

Theoremelpr 3832 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Theoremelpr2 3833 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)

Theoremelpri 3834 If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)

Theoremnelpri 3835 If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)

Theoremelsncg 3836 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremelsnc 3837 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Theoremelsni 3838 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)

Theoremsnidg 3839 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)

Theoremsnidb 3840 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)

Theoremsnid 3841 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)

Theoremelsnc2g 3842 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that , rather than , be a set. (Contributed by NM, 28-Oct-2003.)

Theoremelsnc2 3843 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that , rather than , be a set. (Contributed by NM, 12-Jun-1994.)

Theoremralsns 3844* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremrexsns 3845* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremralsng 3846* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexsng 3847* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)

Theoremexsnrex 3848 There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Theoremralsn 3849* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Theoremrexsn 3850* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremeltpg 3851 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)

Theoremeltpi 3852 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremeltp 3853 A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdftp2 3854* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)

Theoremnfpr 3855 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremifpr 3856 Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)

Theoremralprg 3857* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexprg 3858* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltpg 3859* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextpg 3860* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremralpr 3861* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexpr 3862* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltp 3863* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextp 3864* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremsbcsng 3865* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremnfsn 3866 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)

Theoremcsbsng 3867 Distribute proper substitution through the singleton of a class. csbsng 3867 is derived from the virtual deduction proof csbsngVD 29005. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremdisjsn 3868 Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Theoremdisjsn2 3869 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)

Theoremdisjpr2 3870 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremsnprc 3871 The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)

Theoremr19.12sn 3872* Special case of r19.12 2819 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrabsn 3873* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)

Theoremrabrsn 3874* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.)

Theoremeuabsn2 3875* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremeuabsn 3876 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)

Theoremreusn 3877* A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremabsneu 3878 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)

Theoremrabsneu 3879 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremeusn 3880* Two ways to express " is a singleton." (Contributed by NM, 30-Oct-2010.)

Theoremrabsnt 3881* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Theoremprcom 3882 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Theorempreq1 3883 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)

Theorempreq2 3884 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Theorempreq12 3885 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1i 3886 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2i 3887 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12i 3888 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1d 3889 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2d 3890 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12d 3891 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theoremtpeq1 3892 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq2 3893 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq3 3894 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq1d 3895 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq2d 3896 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq3d 3897 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq123d 3898 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtprot 3899 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpcoma 3900 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)

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