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Theorem List for Metamath Proof Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelrint 3801* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint2 3802* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

2.1.20  Indexed union and intersection

Syntaxciun 3803 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 3727. 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 3804 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 3760. 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 3805* 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 3835. Theorem uniiun 3853 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5625 and funiunfv 5626 are useful when is a function. (Contributed by NM, 27-Jun-1998.)

Definitiondf-iin 3806* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3805. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3836. Theorem intiin 3854 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)

Theoremeliun 3807* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)

Theoremeliin 3808* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)

Theoremiuncom 3809* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Theoremiuncom4 3810 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)

Theoremiunconst 3811* Indexed union of a constant class, i.e. where does not depend on . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinconst 3812* Indexed intersection of a constant class, i.e. where does not depend on . (Contributed by Mario Carneiro, 6-Feb-2015.)

Theoremiuniin 3813* Law combining indexed union with indexed intersection. Eq. 14 in 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 3814* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss1 3815* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)

Theoremiuneq1 3816* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Theoremiineq1 3817* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Theoremss2iun 3818 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2 3819 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2 3820 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2i 3821 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2i 3822 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2d 3823 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)

Theoremiuneq2dv 3824* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)

Theoremiineq2dv 3825* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)

Theoremiuneq1d 3826* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq12d 3827* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq2d 3828* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)

Theoremnfiun 3829 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiin 3830 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiu1 3831 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)

Theoremnfii1 3832 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremdfiun2g 3833* Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremdfiin2g 3834* Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)

Theoremdfiun2 3835* Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremdfiin2 3836* Alternate definition of indexed intersection when is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremcbviun 3837* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Theoremcbviin 3838* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbviunv 3839* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)

Theoremcbviinv 3840* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)

Theoremiunss 3841* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun 3842* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2 3843 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2s 3844* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)

Theoremiunss2 3845* A subclass condition on the members of two indexed classes and that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3756. (Contributed by NM, 9-Dec-2004.)

Theoremiunab 3846* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)

Theoremiunrab 3847* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremiunxdif2 3848* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)

Theoremssiinf 3849 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)

Theoremssiin 3850* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremiinss 3851* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss2 3852 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Theoremuniiun 3853* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)

Theoremintiin 3854* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)

Theoremiunid 3855* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)

Theoremiun0 3856 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorem0iun 3857 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorem0iin 3858 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)

Theoremviin 3859* Indexed intersection with a universal index class. When doesn't depend on , this evaluates to by 19.3 1760 and abid2 2366. When , this evaluates to by intiin 3854 and intv 4080. (Contributed by NM, 11-Sep-2008.)

Theoremiunn0 3860* There is a non-empty class in an indexed collection iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinab 3861* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiinrab 3862* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiinrab2 3863* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiunin2 3864* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3853 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)

Theoremiunin1 3865* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3853 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremiinun2 3866* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3854 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Theoremiundif2 3867* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 3854 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Theorem2iunin 3868* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)

Theoremiindif2 3869* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 3853 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)

Theoremiinin2 3870* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3854 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremiinin1 3871* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3854 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremelriin 3872* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremriin0 3873* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremriinn0 3874* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremriinrab 3875* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremiinxsng 3876* A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiinxprg 3877* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)

Theoremiunxsng 3878* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)

Theoremiunxsn 3879* A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)

Theoremiunun 3880 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiunxun 3881 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiunxiun 3882* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)

Theoremiinuni 3883* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiununi 3884* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremsspwuni 3885 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Theorempwssb 3886* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)

Theoremelpwuni 3887 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Theoremiinpw 3888* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)

Theoremiunpwss 3889* Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)

Theoremrintn0 3890 Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)

2.1.21  Disjointness

Syntaxwdisj 3891 Extend wff notation to include the statement that a family of classes , for , is a disjoint family.
Disj

Definitiondf-disj 3892* A collection of classes is disjoint when for each element , it is in for at most one . (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjss2 3893 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq2 3894 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq2dv 3895* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjss1 3896* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1 3897* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1d 3898* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq12d 3899* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisj 3900* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

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