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

Theoremsnid 3801 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)

Theoremelsnc2g 3802 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 3803 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 3804* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremrexsns 3805* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremralsng 3806* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexsng 3807* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)

Theoremexsnrex 3808 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 3809* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Theoremrexsn 3810* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremeltpg 3811 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)

Theoremeltpi 3812 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremeltp 3813 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 3814* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)

Theoremnfpr 3815 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremifpr 3816 Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)

Theoremralprg 3817* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexprg 3818* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltpg 3819* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextpg 3820* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremralpr 3821* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexpr 3822* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltp 3823* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextp 3824* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremsbcsng 3825* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremnfsn 3826 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)

Theoremcsbsng 3827 Distribute proper substitution through the singleton of a class. csbsng 3827 is derived from the virtual deduction proof csbsngVD 28714. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremdisjsn 3828 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 3829 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)

Theoremdisjpr2 3830 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremsnprc 3831 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 3832* Special case of r19.12 2779 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrabsn 3833* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)

Theoremrabrsn 3834* 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 3835* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremeuabsn 3836 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)

Theoremreusn 3837* 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 3838 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)

Theoremrabsneu 3839 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremeusn 3840* Two ways to express " is a singleton." (Contributed by NM, 30-Oct-2010.)

Theoremrabsnt 3841* 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 3842 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Theorempreq1 3843 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)

Theorempreq2 3844 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Theorempreq12 3845 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1i 3846 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2i 3847 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12i 3848 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1d 3849 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2d 3850 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12d 3851 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theoremtpeq1 3852 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq2 3853 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq3 3854 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq1d 3855 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq2d 3856 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq3d 3857 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq123d 3858 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtprot 3859 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpcoma 3860 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)

Theoremtpcomb 3861 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)

Theoremtpass 3862 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremqdass 3863 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremqdassr 3864 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremtpidm12 3865 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm13 3866 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm23 3867 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm 3868 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtppreq3 3869 An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Theoremprid1g 3870 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid2g 3871 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid1 3872 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprid2 3873 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprprc1 3874 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)

Theoremprprc2 3875 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)

Theoremprprc 3876 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremtpid1 3877 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid2 3878 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid3g 3879 Closed theorem form of tpid3 3880. This proof was automatically generated from the virtual deduction proof tpid3gVD 28663 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpid3 3880 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsnnzg 3881 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)

Theoremsnnz 3882 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremprnz 3883 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Theoremprnzg 3884 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)

Theoremtpnz 3885 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremsnss 3886 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremeldifsn 3887 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)

Theoremeldifsni 3888 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)

Theoremneldifsn 3889 is not in . (Contributed by David Moews, 1-May-2017.)

Theoremneldifsnd 3890 is not in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremrexdifsn 3891 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)

Theoremsnssg 3892 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)

Theoremdifsn 3893 An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremdifprsnss 3894 Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremdifprsn1 3895 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Theoremdifprsn2 3896 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Theoremdiftpsn3 3897 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Theoremtpprceq3 3898 An unordered triple is an unordered pair if one of its elemets is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Theoremtppreqb 3899 An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)

Theoremdifsnb 3900 equals if and only if is not a member of . Generalization of difsn 3893. (Contributed by David Moews, 1-May-2017.)

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