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Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoprcl 3601 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorempwsnss 3602 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)

Theorempwpw0ss 3603 Compute the power set of the power set of the empty set. (See pw0 3539 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 3604 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwtpss 3605 The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwpwpw0ss 3606 Compute the power set of the power set of the power set of the empty set. (See also pw0 3539 and pwpw0ss 3603.) (Contributed by Jim Kingdon, 13-Aug-2018.)

Theorempwv 3607 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 3608 Extend class notation to include the union of a class (read: 'union ')

Definitiondf-uni 3609* 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 2950. (Contributed by NM, 23-Aug-1993.)

Theoremdfuni2 3610* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)

Theoremeluni 3611* Membership in class union. (Contributed by NM, 22-May-1994.)

Theoremeluni2 3612* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)

Theoremelunii 3613 Membership in class union. (Contributed by NM, 24-Mar-1995.)

Theoremnfuni 3614 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfunid 3615 Deduction version of nfuni 3614. (Contributed by NM, 18-Feb-2013.)

Theoremcsbunig 3616 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremunieq 3617 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 3618 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)

Theoremunieqd 3619 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)

Theoremeluniab 3620* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Theoremelunirab 3621* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)

Theoremunipr 3622 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 3623 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 3624 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)

Theoremunisng 3625 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)

Theoremdfnfc2 3626* An alternative statement of the effective freeness of a class , when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremuniun 3627 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)

Theoremuniin 3628 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 3629 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 3630 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremunissi 3631 Subclass relationship for subclass union. Inference form of uniss 3629. (Contributed by David Moews, 1-May-2017.)

Theoremunissd 3632 Subclass relationship for subclass union. Deduction form of uniss 3629. (Contributed by David Moews, 1-May-2017.)

Theoremuni0b 3633 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 3634* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Theoremuni0 3635 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 3636 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 3637 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)

Theoremunissb 3638* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)

Theoremuniss2 3639* 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 3640* 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 3641* Relationship implying union. (Contributed by NM, 10-Nov-1999.)

Theoremunimax 3642* 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 3643 Extend class notation to include the intersection of a class (read: 'intersect ').

Definitiondf-int 3644* 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 2952. (Contributed by NM, 18-Aug-1993.)

Theoremdfint2 3645* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)

Theoreminteq 3646 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)

Theoreminteqi 3647 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)

Theoreminteqd 3648 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)

Theoremelint 3649* Membership in class intersection. (Contributed by NM, 21-May-1994.)

Theoremelint2 3650* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Theoremelintg 3651* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)

Theoremelinti 3652 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremnfint 3653 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremelintab 3654* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)

Theoremelintrab 3655* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)

Theoremelintrabg 3656* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)

Theoremint0 3657 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)

Theoremintss1 3658 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 3659* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)

Theoremssintab 3660* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremssintub 3661* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)

Theoremssmin 3662* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)

Theoremintmin 3663* 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 3664 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)

Theoremintssunim 3665* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)

Theoremssintrab 3666* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)

Theoremintssuni2m 3667* Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)

Theoremintminss 3668* 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 3669* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)

Theoremintmin3 3670* 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 3671* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)

Theoremintab 3672* 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 3673 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)

Theoremintun 3674 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)

Theoremintpr 3675 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)

Theoremintprg 3676 The intersection of a pair is the intersection of its members. Closed form of intpr 3675. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

Theoremintsng 3677 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremintsn 3678 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)

Theoremuniintsnr 3679* The union and intersection of a singleton are equal. See also eusn 3472. (Contributed by Jim Kingdon, 14-Aug-2018.)

Theoremuniintabim 3680 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 3681 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)

Theoremrint0 3682 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint 3683* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint2 3684* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

2.1.20  Indexed union and intersection

Syntaxciun 3685 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 3608. 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 3686 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 3643. 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 3687* 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 distinct variable group (meaning cannot depend on ) and that and do not share a distinct 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 3719. Theorem uniiun 3738 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)

Definitiondf-iin 3688* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3687. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3720. Theorem intiin 3739 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)

Theoremeliun 3689* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)

Theoremeliin 3690* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)

Theoremiuncom 3691* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Theoremiuncom4 3692 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)

Theoremiunconstm 3693* Indexed union of a constant class, i.e. where does not depend on . (Contributed by Jim Kingdon, 15-Aug-2018.)

Theoremiinconstm 3694* Indexed intersection of a constant class, i.e. where does not depend on . (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremiuniin 3695* 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 3696* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss1 3697* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)

Theoremiuneq1 3698* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Theoremiineq1 3699* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Theoremss2iun 3700 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

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