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

Theoremdifsnid 3901 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 3902 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 3903 Compute the power set of the power set of the empty set. (See pw0 3902 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3965, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)

Theoremsnsspr1 3904 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)

Theoremsnsspr2 3905 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)

Theoremsnsstp1 3906 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp2 3907 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp3 3908 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremprss 3909 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 3910 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 3911 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)

Theoremprsspwg 3912 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 3913 Obsolete version of prsspwg 3912 as of 18-Jan-2018. (Contributed by Thierry Arnoux, 3-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsssn 3914 The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)

Theoremssunsn2 3915 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 3968. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremssunsn 3916 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremeqsn 3917* Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.)

Theoremssunpr 3918 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremsspr 3919 The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.)

Theoremsstp 3920 The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremtpss 3921 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.)

Theoremsneqr 3922 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 3923 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Theoremsneqrg 3924 Closed form of sneqr 3922. (Contributed by Scott Fenton, 1-Apr-2011.)

Theoremsneqbg 3925 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremsnsspw 3926 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)

Theoremprsspw 3927 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 3928 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 3929 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 3930 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremprel12 3931 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremopthpr 3932 A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)

Theorempreq12bg 3933 Closed form of preq12b 3930. (Contributed by Scott Fenton, 28-Mar-2014.)

Theoremprneimg 3934 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 3935 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 3936 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Theoremdfopif 3937 Rewrite df-op 3780 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 3938 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdfop 3939 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 3940 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq2 3941 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq12 3942 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)

Theoremopeq1i 3943 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2i 3944 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12i 3945 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theoremopeq1d 3946 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2d 3947 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12d 3948 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremoteq1 3949 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq2 3950 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq3 3951 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq1d 3952 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq2d 3953 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq3d 3954 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq123d 3955 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremnfop 3956 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremnfopd 3957 Deduction version of bound-variable hypothesis builder nfop 3956. This shows how the deduction version of a not-free theorem such as nfop 3956 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)

Theoremopid 3958 The ordered pair in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Theoremralunsn 3959* 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 3960* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremopprc 3961 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopprc1 3962 Expansion of an ordered pair when the first member is a proper class. See also opprc 3961. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopprc2 3963 Expansion of an ordered pair when the second member is a proper class. See also opprc 3961. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremoprcl 3964 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorempwsn 3965 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)

TheorempwsnALT 3966 The power set of a singleton (direct proof). TO DO - should we keep this? (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorempwpr 3967 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)

Theorempwtp 3968 The power set of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theorempwpwpw0 3969 Compute the power set of the power set of the power set of the empty set. (See also pw0 3902 and pwpw0 3903.) (Contributed by NM, 2-May-2009.)

Theorempwv 3970 The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

2.1.18  The union of a class

Syntaxcuni 3971 Extend class notation to include the union of a class (read: 'union ')

Definitiondf-uni 3972* Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For example, (ex-uni 21596). This is similar to the union of two classes df-un 3282. (Contributed by NM, 23-Aug-1993.)

Theoremdfuni2 3973* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)

Theoremeluni 3974* Membership in class union. (Contributed by NM, 22-May-1994.)

Theoremeluni2 3975* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)

Theoremelunii 3976 Membership in class union. (Contributed by NM, 24-Mar-1995.)

Theoremnfuni 3977 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfunid 3978 Deduction version of nfuni 3977. (Contributed by NM, 18-Feb-2013.)

Theoremcsbunig 3979 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremunieq 3980 Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremunieqi 3981 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)

Theoremunieqd 3982 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)

Theoremeluniab 3983* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Theoremelunirab 3984* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)

Theoremunipr 3985 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)

Theoremuniprg 3986 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)

Theoremunisn 3987 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)

Theoremunisng 3988 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)

Theoremdfnfc2 3989* An alternative statement of the effective freeness of a class , when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremuniun 3990 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)

Theoremuniin 3991 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 6934 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremuniss 3992 Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremssuni 3993 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremunissi 3994 Subclass relationship for subclass union. Inference form of uniss 3992. (Contributed by David Moews, 1-May-2017.)

Theoremunissd 3995 Subclass relationship for subclass union. Deduction form of uniss 3992. (Contributed by David Moews, 1-May-2017.)

Theoremuni0b 3996 The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)

Theoremuni0c 3997* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Theoremuni0 3998 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4293 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)

Theoremelssuni 3999 An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)

Theoremunissel 4000 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)

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