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Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsnsspr2 3701 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)

Theoremsnsstp1 3702 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp2 3703 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremsnsstp3 3704 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Theoremprsstp12 3705 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprsstp13 3706 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprsstp23 3707 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprss 3708 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 3709 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 3710 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)

Theoremprsspwg 3711 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.)

Theoremsssnr 3712 Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4158. (Contributed by Jim Kingdon, 10-Aug-2018.)

Theoremsssnm 3713* The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)

Theoremeqsnm 3714* Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremssprr 3715 The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremsstpr 3716 The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremtpss 3717 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 3718 A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremsneqr 3719 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 3720 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Theoremsneqrg 3721 Closed form of sneqr 3719. (Contributed by Scott Fenton, 1-Apr-2011.)

Theoremsneqbg 3722 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremsnsspw 3723 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)

Theoremprsspw 3724 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.)

Theorempreqr1g 3725 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3727. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theorempreqr2g 3726 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3728. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theorempreqr1 3727 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 3728 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 3729 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremprel12 3730 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)

Theoremopthpr 3731 A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)

Theorempreq12bg 3732 Closed form of preq12b 3729. (Contributed by Scott Fenton, 28-Mar-2014.)

Theoremprneimg 3733 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.)

Theorempreqsn 3734 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Theoremdfopg 3735 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdfop 3736 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 3737 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq2 3738 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeq12 3739 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)

Theoremopeq1i 3740 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2i 3741 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12i 3742 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theoremopeq1d 3743 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq2d 3744 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Theoremopeq12d 3745 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremoteq1 3746 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq2 3747 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq3 3748 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Theoremoteq1d 3749 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq2d 3750 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq3d 3751 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremoteq123d 3752 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Theoremnfop 3753 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremnfopd 3754 Deduction version of bound-variable hypothesis builder nfop 3753. This shows how the deduction version of a not-free theorem such as nfop 3753 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)

Theoremopid 3755 The ordered pair in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Theoremralunsn 3756* 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 3757* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremopprc 3758 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopprc1 3759 Expansion of an ordered pair when the first member is a proper class. See also opprc 3758. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopprc2 3760 Expansion of an ordered pair when the second member is a proper class. See also opprc 3758. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremoprcl 3761 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorempwsnss 3762 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)

Theorempwpw0ss 3763 Compute the power set of the power set of the empty set. (See pw0 3699 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with subset in place of equality). (Contributed by Jim Kingdon, 12-Aug-2018.)

Theorempwprss 3764 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwtpss 3765 The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwpwpw0ss 3766 Compute the power set of the power set of the power set of the empty set. (See also pw0 3699 and pwpw0ss 3763.) (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwv 3767 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 3768 Extend class notation to include the union of a class. Read: "union (of) ".

Definitiondf-uni 3769* 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, . This is similar to the union of two classes df-un 3102. (Contributed by NM, 23-Aug-1993.)

Theoremdfuni2 3770* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)

Theoremeluni 3771* Membership in class union. (Contributed by NM, 22-May-1994.)

Theoremeluni2 3772* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)

Theoremelunii 3773 Membership in class union. (Contributed by NM, 24-Mar-1995.)

Theoremnfuni 3774 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfunid 3775 Deduction version of nfuni 3774. (Contributed by NM, 18-Feb-2013.)

Theoremcsbunig 3776 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremunieq 3777 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 3778 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)

Theoremunieqd 3779 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)

Theoremeluniab 3780* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Theoremelunirab 3781* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)

Theoremunipr 3782 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 3783 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 3784 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)

Theoremunisng 3785 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)

Theoremdfnfc2 3786* An alternate statement of the effective freeness of a class , when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremuniun 3787 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)

Theoremuniin 3788 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremuniss 3789 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 3790 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremunissi 3791 Subclass relationship for subclass union. Inference form of uniss 3789. (Contributed by David Moews, 1-May-2017.)

Theoremunissd 3792 Subclass relationship for subclass union. Deduction form of uniss 3789. (Contributed by David Moews, 1-May-2017.)

Theoremuni0b 3793 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 3794* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Theoremuni0 3795 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)

Theoremelssuni 3796 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 3797 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)

Theoremunissb 3798* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)

Theoremuniss2 3799* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)

Theoremunidif 3800* If the difference contains the largest members of , then the union of the difference is the union of . (Contributed by NM, 22-Mar-2004.)

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