Type  Label  Description 
Statement 

Theorem  elint2 3701* 
Membership in class intersection. (Contributed by NM, 14Oct1999.)



Theorem  elintg 3702* 
Membership in class intersection, with the sethood requirement expressed
as an antecedent. (Contributed by NM, 20Nov2003.)



Theorem  elinti 3703 
Membership in class intersection. (Contributed by NM, 14Oct1999.)
(Proof shortened by Andrew Salmon, 9Jul2011.)



Theorem  nfint 3704 
Boundvariable hypothesis builder for intersection. (Contributed by NM,
2Feb1997.) (Proof shortened by Andrew Salmon, 12Aug2011.)



Theorem  elintab 3705* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 30Aug1993.)



Theorem  elintrab 3706* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 17Oct1999.)



Theorem  elintrabg 3707* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 17Feb2007.)



Theorem  int0 3708 
The intersection of the empty set is the universal class. Exercise 2 of
[TakeutiZaring] p. 44.
(Contributed by NM, 18Aug1993.)



Theorem  intss1 3709 
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, 18Nov1995.)



Theorem  ssint 3710* 
Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
and its converse. (Contributed by NM, 14Oct1999.)



Theorem  ssintab 3711* 
Subclass of the intersection of a class abstraction. (Contributed by
NM, 31Jul2006.) (Proof shortened by Andrew Salmon, 9Jul2011.)



Theorem  ssintub 3712* 
Subclass of the least upper bound. (Contributed by NM, 8Aug2000.)



Theorem  ssmin 3713* 
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10Aug2006.)



Theorem  intmin 3714* 
Any member of a class is the smallest of those members that include it.
(Contributed by NM, 13Aug2002.) (Proof shortened by Andrew Salmon,
9Jul2011.)



Theorem  intss 3715 
Intersection of subclasses. (Contributed by NM, 14Oct1999.)



Theorem  intssunim 3716* 
The intersection of an inhabited set is a subclass of its union.
(Contributed by NM, 29Jul2006.)



Theorem  ssintrab 3717* 
Subclass of the intersection of a restricted class builder.
(Contributed by NM, 30Jan2015.)



Theorem  intssuni2m 3718* 
Subclass relationship for intersection and union. (Contributed by Jim
Kingdon, 14Aug2018.)



Theorem  intminss 3719* 
Under subset ordering, the intersection of a restricted class
abstraction is less than or equal to any of its members. (Contributed
by NM, 7Sep2013.)



Theorem  intmin2 3720* 
Any set is the smallest of all sets that include it. (Contributed by
NM, 20Sep2003.)



Theorem  intmin3 3721* 
Under subset ordering, the intersection of a class abstraction is less
than or equal to any of its members. (Contributed by NM,
3Jul2005.)



Theorem  intmin4 3722* 
Elimination of a conjunct in a class intersection. (Contributed by NM,
31Jul2006.)



Theorem  intab 3723* 
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, 28Jul2006.) (Proof shortened by
Mario Carneiro, 14Nov2016.)



Theorem  int0el 3724 
The intersection of a class containing the empty set is empty.
(Contributed by NM, 24Apr2004.)



Theorem  intun 3725 
The class intersection of the union of two classes. Theorem 78 of
[Suppes] p. 42. (Contributed by NM,
22Sep2002.)



Theorem  intpr 3726 
The intersection of a pair is the intersection of its members. Theorem
71 of [Suppes] p. 42. (Contributed by
NM, 14Oct1999.)



Theorem  intprg 3727 
The intersection of a pair is the intersection of its members. Closed
form of intpr 3726. Theorem 71 of [Suppes] p. 42. (Contributed by FL,
27Apr2008.)



Theorem  intsng 3728 
Intersection of a singleton. (Contributed by Stefan O'Rear,
22Feb2015.)



Theorem  intsn 3729 
The intersection of a singleton is its member. Theorem 70 of [Suppes]
p. 41. (Contributed by NM, 29Sep2002.)



Theorem  uniintsnr 3730* 
The union and intersection of a singleton are equal. See also eusn 3520.
(Contributed by Jim Kingdon, 14Aug2018.)



Theorem  uniintabim 3731 
The union and the intersection of a class abstraction are equal if there
is a unique satisfying value of . (Contributed by Jim
Kingdon, 14Aug2018.)



Theorem  intunsn 3732 
Theorem joining a singleton to an intersection. (Contributed by NM,
29Sep2002.)



Theorem  rint0 3733 
Relative intersection of an empty set. (Contributed by Stefan O'Rear,
3Apr2015.)



Theorem  elrint 3734* 
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3Apr2015.)



Theorem  elrint2 3735* 
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3Apr2015.)



2.1.20 Indexed union and
intersection


Syntax  ciun 3736 
Extend class notation to include indexed union. Note: Historically
(prior to 21Oct2005), set.mm used the notation
, with
the same union symbol as cuni 3659. 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.



Syntax  ciin 3737 
Extend class notation to include indexed intersection. Note:
Historically (prior to 21Oct2005), set.mm used the notation
, with the
same intersection symbol as cint 3694. 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.



Definition  dfiun 3738* 
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 3770. Theorem uniiun 3789 provides
a definition of ordinary union in terms of indexed union. (Contributed
by NM, 27Jun1998.)



Definition  dfiin 3739* 
Define indexed intersection. Definition of [Stoll] p. 45. See the
remarks for its sibling operation of indexed union dfiun 3738. An
alternate definition tying indexed intersection to ordinary intersection
is dfiin2 3771. Theorem intiin 3790 provides a definition of ordinary
intersection in terms of indexed intersection. (Contributed by NM,
27Jun1998.)



Theorem  eliun 3740* 
Membership in indexed union. (Contributed by NM, 3Sep2003.)



Theorem  eliin 3741* 
Membership in indexed intersection. (Contributed by NM, 3Sep2003.)



Theorem  iuncom 3742* 
Commutation of indexed unions. (Contributed by NM, 18Dec2008.)



Theorem  iuncom4 3743 
Commutation of union with indexed union. (Contributed by Mario
Carneiro, 18Jan2014.)



Theorem  iunconstm 3744* 
Indexed union of a constant class, i.e. where does not depend on
. (Contributed
by Jim Kingdon, 15Aug2018.)



Theorem  iinconstm 3745* 
Indexed intersection of a constant class, i.e. where does not
depend on .
(Contributed by Jim Kingdon, 19Dec2018.)



Theorem  iuniin 3746* 
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, 17Aug2004.) (Proof shortened by Andrew Salmon,
25Jul2011.)



Theorem  iunss1 3747* 
Subclass theorem for indexed union. (Contributed by NM, 10Dec2004.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iinss1 3748* 
Subclass theorem for indexed union. (Contributed by NM,
24Jan2012.)



Theorem  iuneq1 3749* 
Equality theorem for indexed union. (Contributed by NM,
27Jun1998.)



Theorem  iineq1 3750* 
Equality theorem for restricted existential quantifier. (Contributed by
NM, 27Jun1998.)



Theorem  ss2iun 3751 
Subclass theorem for indexed union. (Contributed by NM, 26Nov2003.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iuneq2 3752 
Equality theorem for indexed union. (Contributed by NM,
22Oct2003.)



Theorem  iineq2 3753 
Equality theorem for indexed intersection. (Contributed by NM,
22Oct2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iuneq2i 3754 
Equality inference for indexed union. (Contributed by NM,
22Oct2003.)



Theorem  iineq2i 3755 
Equality inference for indexed intersection. (Contributed by NM,
22Oct2003.)



Theorem  iineq2d 3756 
Equality deduction for indexed intersection. (Contributed by NM,
7Dec2011.)



Theorem  iuneq2dv 3757* 
Equality deduction for indexed union. (Contributed by NM,
3Aug2004.)



Theorem  iineq2dv 3758* 
Equality deduction for indexed intersection. (Contributed by NM,
3Aug2004.)



Theorem  iuneq1d 3759* 
Equality theorem for indexed union, deduction version. (Contributed by
Drahflow, 22Oct2015.)



Theorem  iuneq12d 3760* 
Equality deduction for indexed union, deduction version. (Contributed
by Drahflow, 22Oct2015.)



Theorem  iuneq2d 3761* 
Equality deduction for indexed union. (Contributed by Drahflow,
22Oct2015.)



Theorem  nfiunxy 3762* 
Boundvariable hypothesis builder for indexed union. (Contributed by
Mario Carneiro, 25Jan2014.)



Theorem  nfiinxy 3763* 
Boundvariable hypothesis builder for indexed intersection.
(Contributed by Mario Carneiro, 25Jan2014.)



Theorem  nfiunya 3764* 
Boundvariable hypothesis builder for indexed union. (Contributed by
Mario Carneiro, 25Jan2014.)



Theorem  nfiinya 3765* 
Boundvariable hypothesis builder for indexed intersection.
(Contributed by Mario Carneiro, 25Jan2014.)



Theorem  nfiu1 3766 
Boundvariable hypothesis builder for indexed union. (Contributed by
NM, 12Oct2003.)



Theorem  nfii1 3767 
Boundvariable hypothesis builder for indexed intersection.
(Contributed by NM, 15Oct2003.)



Theorem  dfiun2g 3768* 
Alternate definition of indexed union when is a set. Definition
15(a) of [Suppes] p. 44. (Contributed by
NM, 23Mar2006.) (Proof
shortened by Andrew Salmon, 25Jul2011.)



Theorem  dfiin2g 3769* 
Alternate definition of indexed intersection when is a set.
(Contributed by Jeff Hankins, 27Aug2009.)



Theorem  dfiun2 3770* 
Alternate definition of indexed union when is a set. Definition
15(a) of [Suppes] p. 44. (Contributed by
NM, 27Jun1998.) (Revised by
David Abernethy, 19Jun2012.)



Theorem  dfiin2 3771* 
Alternate definition of indexed intersection when is a set.
Definition 15(b) of [Suppes] p. 44.
(Contributed by NM, 28Jun1998.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  dfiunv2 3772* 
Define double indexed union. (Contributed by FL, 6Nov2013.)



Theorem  cbviun 3773* 
Rule used to change the bound variables in an indexed union, with the
substitution specified implicitly by the hypothesis. (Contributed by
NM, 26Mar2006.) (Revised by Andrew Salmon, 25Jul2011.)



Theorem  cbviin 3774* 
Change bound variables in an indexed intersection. (Contributed by Jeff
Hankins, 26Aug2009.) (Revised by Mario Carneiro, 14Oct2016.)



Theorem  cbviunv 3775* 
Rule used to change the bound variables in an indexed union, with the
substitution specified implicitly by the hypothesis. (Contributed by
NM, 15Sep2003.)



Theorem  cbviinv 3776* 
Change bound variables in an indexed intersection. (Contributed by Jeff
Hankins, 26Aug2009.)



Theorem  iunss 3777* 
Subset theorem for an indexed union. (Contributed by NM, 13Sep2003.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  ssiun 3778* 
Subset implication for an indexed union. (Contributed by NM,
3Sep2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  ssiun2 3779 
Identity law for subset of an indexed union. (Contributed by NM,
12Oct2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  ssiun2s 3780* 
Subset relationship for an indexed union. (Contributed by NM,
26Oct2003.)



Theorem  iunss2 3781* 
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 3690. (Contributed by NM, 9Dec2004.)



Theorem  iunab 3782* 
The indexed union of a class abstraction. (Contributed by NM,
27Dec2004.)



Theorem  iunrab 3783* 
The indexed union of a restricted class abstraction. (Contributed by
NM, 3Jan2004.) (Proof shortened by Mario Carneiro, 14Nov2016.)



Theorem  iunxdif2 3784* 
Indexed union with a class difference as its index. (Contributed by NM,
10Dec2004.)



Theorem  ssiinf 3785 
Subset theorem for an indexed intersection. (Contributed by FL,
15Oct2012.) (Proof shortened by Mario Carneiro, 14Oct2016.)



Theorem  ssiin 3786* 
Subset theorem for an indexed intersection. (Contributed by NM,
15Oct2003.)



Theorem  iinss 3787* 
Subset implication for an indexed intersection. (Contributed by NM,
15Oct2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iinss2 3788 
An indexed intersection is included in any of its members. (Contributed
by FL, 15Oct2012.)



Theorem  uniiun 3789* 
Class union in terms of indexed union. Definition in [Stoll] p. 43.
(Contributed by NM, 28Jun1998.)



Theorem  intiin 3790* 
Class intersection in terms of indexed intersection. Definition in
[Stoll] p. 44. (Contributed by NM,
28Jun1998.)



Theorem  iunid 3791* 
An indexed union of singletons recovers the index set. (Contributed by
NM, 6Sep2005.)



Theorem  iun0 3792 
An indexed union of the empty set is empty. (Contributed by NM,
26Mar2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  0iun 3793 
An empty indexed union is empty. (Contributed by NM, 4Dec2004.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  0iin 3794 
An empty indexed intersection is the universal class. (Contributed by
NM, 20Oct2005.)



Theorem  viin 3795* 
Indexed intersection with a universal index class. (Contributed by NM,
11Sep2008.)



Theorem  iunn0m 3796* 
There is an inhabited class in an indexed collection iff the
indexed union of them is inhabited. (Contributed by Jim Kingdon,
16Aug2018.)



Theorem  iinab 3797* 
Indexed intersection of a class builder. (Contributed by NM,
6Dec2011.)



Theorem  iinrabm 3798* 
Indexed intersection of a restricted class builder. (Contributed by Jim
Kingdon, 16Aug2018.)



Theorem  iunin2 3799* 
Indexed union of intersection. Generalization of half of theorem
"Distributive laws" in [Enderton] p. 30. Use uniiun 3789 to recover
Enderton's theorem. (Contributed by NM, 26Mar2004.)



Theorem  iunin1 3800* 
Indexed union of intersection. Generalization of half of theorem
"Distributive laws" in [Enderton] p. 30. Use uniiun 3789 to recover
Enderton's theorem. (Contributed by Mario Carneiro, 30Aug2015.)

