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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremralsn 4001* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Theoremrexsn 4002* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremelpwunsn 4003 Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)

Theoremeqoreldif 4004 An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.)

Theoremeltpg 4005 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)

Theoremeldiftp 4006 Membership in a set with three elements removed. Similar to eldifsn 4088 and eldifpr 3982. (Contributed by David A. Wheeler, 22-Jul-2017.)

Theoremeltpi 4007 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremeltp 4008 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 4009* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)

Theoremnfpr 4010 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremifpr 4011 Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)

Theoremralprg 4012* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexprg 4013* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltpg 4014* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextpg 4015* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremralpr 4016* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexpr 4017* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltp 4018* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextp 4019* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremnfsn 4020 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)

Theoremcsbsng 4021 Distribute proper substitution through the singleton of a class. csbsng 4021 is derived from the virtual deduction proof csbsngVD 37353. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremcsbprg 4022 Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)

Theoremdisjsn 4023 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 4024 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)

Theoremdisjpr2 4025 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremdisjprsn 4026 The disjoint intersection of an unordered pair and a singleton. (Contributed by AV, 23-Jan-2021.)

Theoremsnprc 4027 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, 21-Jun-1993.)

Theoremsnnzb 4028 A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)

Theoremr19.12sn 4029* Special case of r19.12 2903 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)

Theoremrabsn 4030* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)

Theoremrabsnifsb 4031* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)

Theoremrabsnif 4032* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)

Theoremrabrsn 4033* 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.) (Proof shortened by AV, 21-Jul-2019.)

Theoremeuabsn2 4034* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremeuabsn 4035 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)

Theoremreusn 4036* 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 4037 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)

Theoremrabsneu 4038 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremeusn 4039* Two ways to express " is a singleton." (Contributed by NM, 30-Oct-2010.)

Theoremrabsnt 4040* 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 4041 Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.)

Theorempreq1 4042 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)

Theorempreq2 4043 Equality theorem for unordered pairs. (Contributed by NM, 15-Jul-1993.)

Theorempreq12 4044 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1i 4045 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2i 4046 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12i 4047 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1d 4048 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2d 4049 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12d 4050 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theoremtpeq1 4051 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq2 4052 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq3 4053 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq1d 4054 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq2d 4055 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq3d 4056 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq123d 4057 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtprot 4058 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpcoma 4059 Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.)

Theoremtpcomb 4060 Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.)

Theoremtpass 4061 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremqdass 4062 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremqdassr 4063 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremtpidm12 4064 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm13 4065 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm23 4066 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm 4067 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtppreq3 4068 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 4069 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid2g 4070 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid1 4071 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 24-Jun-1993.)

Theoremprid2 4072 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprprc1 4073 A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)

Theoremprprc2 4074 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)

Theoremprprc 4075 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremtpid1 4076 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid2 4077 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid3g 4078 Closed theorem form of tpid3 4079. This proof was automatically generated from the virtual deduction proof tpid3gVD 37301 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpid3 4079 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsnnzg 4080 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)

Theoremsnnz 4081 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremprnz 4082 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Theoremprnzg 4083 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)

Theoremtpnz 4084 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremtpnzd 4085 A triplet containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.)

Theoremraltpd 4086* Convert a quantification over a triple to a conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)

Theoremsnss 4087 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.)

Theoremeldifsn 4088 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)

Theoremeldifsni 4089 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)

Theoremneldifsn 4090 is not in . (Contributed by David Moews, 1-May-2017.)

Theoremneldifsnd 4091 is not in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremrexdifsn 4092 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)

Theoremraldifsni 4093 Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremraldifsnb 4094* Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.)

Theoremeldifvsn 4095 A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)

Theoremsnssg 4096 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 4097 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 4098 Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremdifprsn1 4099 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Theoremdifprsn2 4100 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

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