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

Theorempwtpss 3701 The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwpwpw0ss 3702 Compute the power set of the power set of the power set of the empty set. (See also pw0 3635 and pwpw0ss 3699.) (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwv 3703 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 3704 Extend class notation to include the union of a class (read: 'union ')

Definitiondf-uni 3705* 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, { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } . This is similar to the union of two classes df-un 3043. (Contributed by NM, 23-Aug-1993.)

Theoremdfuni2 3706* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)

Theoremeluni 3707* Membership in class union. (Contributed by NM, 22-May-1994.)

Theoremeluni2 3708* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)

Theoremelunii 3709 Membership in class union. (Contributed by NM, 24-Mar-1995.)

Theoremnfuni 3710 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfunid 3711 Deduction version of nfuni 3710. (Contributed by NM, 18-Feb-2013.)

Theoremcsbunig 3712 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremunieq 3713 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 3714 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)

Theoremunieqd 3715 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)

Theoremeluniab 3716* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Theoremelunirab 3717* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)

Theoremunipr 3718 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 3719 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 3720 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)

Theoremunisng 3721 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)

Theoremdfnfc2 3722* An alternate statement of the effective freeness of a class , when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremuniun 3723 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)

Theoremuniin 3724 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 3725 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 3726 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremunissi 3727 Subclass relationship for subclass union. Inference form of uniss 3725. (Contributed by David Moews, 1-May-2017.)

Theoremunissd 3728 Subclass relationship for subclass union. Deduction form of uniss 3725. (Contributed by David Moews, 1-May-2017.)

Theoremuni0b 3729 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 3730* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Theoremuni0 3731 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 3732 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 3733 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)

Theoremunissb 3734* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)

Theoremuniss2 3735* 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 3736* If the difference contains the largest members of , then the union of the difference is the union of . (Contributed by NM, 22-Mar-2004.)

Theoremssunieq 3737* Relationship implying union. (Contributed by NM, 10-Nov-1999.)

Theoremunimax 3738* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)

2.1.19  The intersection of a class

Syntaxcint 3739 Extend class notation to include the intersection of a class (read: 'intersect ').

Definitiondf-int 3740* Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example, { { 1 , 3 } , { 1 , 8 } } = { 1 } . Compare this with the intersection of two classes, df-in 3045. (Contributed by NM, 18-Aug-1993.)

Theoremdfint2 3741* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)

Theoreminteq 3742 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)

Theoreminteqi 3743 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)

Theoreminteqd 3744 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)

Theoremelint 3745* Membership in class intersection. (Contributed by NM, 21-May-1994.)

Theoremelint2 3746* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Theoremelintg 3747* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)

Theoremelinti 3748 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremnfint 3749 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremelintab 3750* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)

Theoremelintrab 3751* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)

Theoremelintrabg 3752* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)

Theoremint0 3753 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)

Theoremintss1 3754 An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)

Theoremssint 3755* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)

Theoremssintab 3756* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremssintub 3757* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)

Theoremssmin 3758* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)

Theoremintmin 3759* Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremintss 3760 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)

Theoremintssunim 3761* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)

Theoremssintrab 3762* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)

Theoremintssuni2m 3763* Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)

Theoremintminss 3764* Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)

Theoremintmin2 3765* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)

Theoremintmin3 3766* Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)

Theoremintmin4 3767* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)

Theoremintab 3768* The intersection of a special case of a class abstraction. may be free in and , which can be thought of a and . (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremint0el 3769 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)

Theoremintun 3770 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)

Theoremintpr 3771 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)

Theoremintprg 3772 The intersection of a pair is the intersection of its members. Closed form of intpr 3771. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

Theoremintsng 3773 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremintsn 3774 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)

Theoremuniintsnr 3775* The union and intersection of a singleton are equal. See also eusn 3565. (Contributed by Jim Kingdon, 14-Aug-2018.)

Theoremuniintabim 3776 The union and the intersection of a class abstraction are equal if there is a unique satisfying value of . (Contributed by Jim Kingdon, 14-Aug-2018.)

Theoremintunsn 3777 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)

Theoremrint0 3778 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint 3779* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint2 3780* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

2.1.20  Indexed union and intersection

Syntaxciun 3781 Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation , with the same union symbol as cuni 3704. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses as distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.

Syntaxciin 3782 Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation , with the same intersection symbol as cint 3739. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.

Definitiondf-iun 3783* Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, is independent of (although this is not required by the definition), and depends on i.e. can be read informally as . We call the index, the index set, and the indexed set. In most books, is written as a subscript or underneath a union symbol . We use a special union symbol to make it easier to distinguish from plain class union. In many theorems, you will see that and are in the same disjoint variable group (meaning cannot depend on ) and that and do not share a disjoint variable group (meaning that can be thought of as i.e. can be substituted with a class expression containing ). An alternate definition tying indexed union to ordinary union is dfiun2 3815. Theorem uniiun 3834 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)

Definitiondf-iin 3784* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3783. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3816. Theorem intiin 3835 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)

Theoremeliun 3785* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)

Theoremeliin 3786* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)

Theoremiuncom 3787* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Theoremiuncom4 3788 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)

Theoremiunconstm 3789* Indexed union of a constant class, i.e. where does not depend on . (Contributed by Jim Kingdon, 15-Aug-2018.)

Theoremiinconstm 3790* Indexed intersection of a constant class, i.e. where does not depend on . (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremiuniin 3791* Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiunss1 3792* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss1 3793* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)

Theoremiuneq1 3794* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Theoremiineq1 3795* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Theoremss2iun 3796 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2 3797 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2 3798 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2i 3799 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2i 3800 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)

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