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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtppreq3 3901 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 3902 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid2g 3903 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid1 3904 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprid2 3905 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprprc1 3906 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)

Theoremprprc2 3907 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)

Theoremprprc 3908 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremtpid1 3909 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid2 3910 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid3g 3911 Closed theorem form of tpid3 3912. This proof was automatically generated from the virtual deduction proof tpid3gVD 28808 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpid3 3912 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsnnzg 3913 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)

Theoremsnnz 3914 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremprnz 3915 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Theoremprnzg 3916 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)

Theoremtpnz 3917 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremsnss 3918 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 3919 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)

Theoremeldifsni 3920 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)

Theoremneldifsn 3921 is not in . (Contributed by David Moews, 1-May-2017.)

Theoremneldifsnd 3922 is not in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremrexdifsn 3923 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)

Theoremsnssg 3924 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 3925 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 3926 Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremdifprsn1 3927 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Theoremdifprsn2 3928 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Theoremdiftpsn3 3929 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Theoremtpprceq3 3930 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 3931 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 3932 equals if and only if is not a member of . Generalization of difsn 3925. (Contributed by David Moews, 1-May-2017.)

Theoremdifsnpss 3933 is a proper subclass of if and only if is a member of . (Contributed by David Moews, 1-May-2017.)

Theoremsnssi 3934 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)

Theoremsnssd 3935 The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremdifsnid 3936 If we remove a single element from a class then put it back in, we end up with the original class. (Contributed by NM, 2-Oct-2006.)

Theorempw0 3937 Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theorempwpw0 3938 Compute the power set of the power set of the empty set. (See pw0 3937 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4001, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)

Theoremsnsspr1 3939 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)

Theoremsnsspr2 3940 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)

Theoremsnsstp1 3941 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp2 3942 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp3 3943 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremprss 3944 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremprssg 3945 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremprssi 3946 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)

Theoremprsspwg 3947 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)

TheoremprsspwgOLD 3948 Obsolete version of prsspwg 3947 as of 18-Jan-2018. (Contributed by Thierry Arnoux, 3-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsssn 3949 The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)

Theoremssunsn2 3950 The property of being sandwiched between two sets naturally splits under union with a singleton. This is the induction hypothesis for the determination of large powersets such as pwtp 4004. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremssunsn 3951 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremeqsn 3952* Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.)

Theoremssunpr 3953 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremsspr 3954 The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.)

Theoremsstp 3955 The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremtpss 3956 A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpssi 3957 A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremsneqr 3958 If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)

Theoremsnsssn 3959 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Theoremsneqrg 3960 Closed form of sneqr 3958. (Contributed by Scott Fenton, 1-Apr-2011.)

Theoremsneqbg 3961 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremsnsspw 3962 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)

Theoremprsspw 3963 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempreqr1 3964 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)

Theorempreqr2 3965 Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)

Theorempreq12b 3966 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremprel12 3967 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremopthpr 3968 A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)

Theorempreq12bg 3969 Closed form of preq12b 3966. (Contributed by Scott Fenton, 28-Mar-2014.)

Theoremprneimg 3970 Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)

Theoremprnebg 3971 A (proper) pair is not equal to another (maybe inproper) pair if and only if an element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 16-Jan-2018.)

Theorempreqsn 3972 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Theoremdfopif 3973 Rewrite df-op 3815 using . When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdfopg 3974 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdfop 3975 Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)

Theoremopeq1 3976 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq2 3977 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq12 3978 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)

Theoremopeq1i 3979 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2i 3980 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12i 3981 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theoremopeq1d 3982 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2d 3983 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12d 3984 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremoteq1 3985 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq2 3986 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq3 3987 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq1d 3988 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq2d 3989 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq3d 3990 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq123d 3991 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremnfop 3992 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremnfopd 3993 Deduction version of bound-variable hypothesis builder nfop 3992. This shows how the deduction version of a not-free theorem such as nfop 3992 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)

Theoremopid 3994 The ordered pair in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Theoremralunsn 3995* Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theorem2ralunsn 3996* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremopprc 3997 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopprc1 3998 Expansion of an ordered pair when the first member is a proper class. See also opprc 3997. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopprc2 3999 Expansion of an ordered pair when the second member is a proper class. See also opprc 3997. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremoprcl 4000 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

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