Type  Label  Description 
Statement 

Theorem  pwtpss 3701 
The power set of an unordered triple. (Contributed by Jim Kingdon,
13Aug2018.)



Theorem  pwpwpw0ss 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,
13Aug2018.)



Theorem  pwv 3703 
The power class of the universe is the universe. Exercise 4.12(d) of
[Mendelson] p. 235. (Contributed by NM,
14Sep2003.)



2.1.18 The union of a class


Syntax  cuni 3704 
Extend class notation to include the union of a class (read: 'union
')



Definition  dfuni 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 dfun 3043. (Contributed by NM, 23Aug1993.)



Theorem  dfuni2 3706* 
Alternate definition of class union. (Contributed by NM,
28Jun1998.)



Theorem  eluni 3707* 
Membership in class union. (Contributed by NM, 22May1994.)



Theorem  eluni2 3708* 
Membership in class union. Restricted quantifier version. (Contributed
by NM, 31Aug1999.)



Theorem  elunii 3709 
Membership in class union. (Contributed by NM, 24Mar1995.)



Theorem  nfuni 3710 
Boundvariable hypothesis builder for union. (Contributed by NM,
30Dec1996.) (Proof shortened by Andrew Salmon, 27Aug2011.)



Theorem  nfunid 3711 
Deduction version of nfuni 3710. (Contributed by NM, 18Feb2013.)



Theorem  csbunig 3712 
Distribute proper substitution through the union of a class.
(Contributed by Alan Sare, 10Nov2012.)



Theorem  unieq 3713 
Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.
(Contributed by NM, 10Aug1993.) (Proof shortened by Andrew Salmon,
29Jun2011.)



Theorem  unieqi 3714 
Inference of equality of two class unions. (Contributed by NM,
30Aug1993.)



Theorem  unieqd 3715 
Deduction of equality of two class unions. (Contributed by NM,
21Apr1995.)



Theorem  eluniab 3716* 
Membership in union of a class abstraction. (Contributed by NM,
11Aug1994.) (Revised by Mario Carneiro, 14Nov2016.)



Theorem  elunirab 3717* 
Membership in union of a class abstraction. (Contributed by NM,
4Oct2006.)



Theorem  unipr 3718 
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 23Aug1993.)



Theorem  uniprg 3719 
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 25Aug2006.)



Theorem  unisn 3720 
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 30Aug1993.)



Theorem  unisng 3721 
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 13Aug2002.)



Theorem  dfnfc2 3722* 
An alternate statement of the effective freeness of a class , when
it is a set. (Contributed by Mario Carneiro, 14Oct2016.)



Theorem  uniun 3723 
The class union of the union of two classes. Theorem 8.3 of [Quine]
p. 53. (Contributed by NM, 20Aug1993.)



Theorem  uniin 3724 
The class union of the intersection of two classes. Exercise 4.12(n) of
[Mendelson] p. 235. (Contributed by
NM, 4Dec2003.) (Proof shortened
by Andrew Salmon, 29Jun2011.)



Theorem  uniss 3725 
Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
(Contributed by NM, 22Mar1998.) (Proof shortened by Andrew Salmon,
29Jun2011.)



Theorem  ssuni 3726 
Subclass relationship for class union. (Contributed by NM,
24May1994.) (Proof shortened by Andrew Salmon, 29Jun2011.)



Theorem  unissi 3727 
Subclass relationship for subclass union. Inference form of uniss 3725.
(Contributed by David Moews, 1May2017.)



Theorem  unissd 3728 
Subclass relationship for subclass union. Deduction form of uniss 3725.
(Contributed by David Moews, 1May2017.)



Theorem  uni0b 3729 
The union of a set is empty iff the set is included in the singleton of
the empty set. (Contributed by NM, 12Sep2004.)



Theorem  uni0c 3730* 
The union of a set is empty iff all of its members are empty.
(Contributed by NM, 16Aug2006.)



Theorem  uni0 3731 
The union of the empty set is the empty set. Theorem 8.7 of [Quine]
p. 54. (Reproved without relying on axnul by Eric Schmidt.)
(Contributed by NM, 16Sep1993.) (Revised by Eric Schmidt,
4Apr2007.)



Theorem  elssuni 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, 6Jun1994.)



Theorem  unissel 3733 
Condition turning a subclass relationship for union into an equality.
(Contributed by NM, 18Jul2006.)



Theorem  unissb 3734* 
Relationship involving membership, subset, and union. Exercise 5 of
[Enderton] p. 26 and its converse.
(Contributed by NM, 20Sep2003.)



Theorem  uniss2 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, 22Mar2004.)



Theorem  unidif 3736* 
If the difference
contains the largest
members of , then
the union of the difference is the union of . (Contributed by NM,
22Mar2004.)



Theorem  ssunieq 3737* 
Relationship implying union. (Contributed by NM, 10Nov1999.)



Theorem  unimax 3738* 
Any member of a class is the largest of those members that it includes.
(Contributed by NM, 13Aug2002.)



2.1.19 The intersection of a class


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



Definition  dfint 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, dfin 3045.
(Contributed by NM, 18Aug1993.)



Theorem  dfint2 3741* 
Alternate definition of class intersection. (Contributed by NM,
28Jun1998.)



Theorem  inteq 3742 
Equality law for intersection. (Contributed by NM, 13Sep1999.)



Theorem  inteqi 3743 
Equality inference for class intersection. (Contributed by NM,
2Sep2003.)



Theorem  inteqd 3744 
Equality deduction for class intersection. (Contributed by NM,
2Sep2003.)



Theorem  elint 3745* 
Membership in class intersection. (Contributed by NM, 21May1994.)



Theorem  elint2 3746* 
Membership in class intersection. (Contributed by NM, 14Oct1999.)



Theorem  elintg 3747* 
Membership in class intersection, with the sethood requirement expressed
as an antecedent. (Contributed by NM, 20Nov2003.)



Theorem  elinti 3748 
Membership in class intersection. (Contributed by NM, 14Oct1999.)
(Proof shortened by Andrew Salmon, 9Jul2011.)



Theorem  nfint 3749 
Boundvariable hypothesis builder for intersection. (Contributed by NM,
2Feb1997.) (Proof shortened by Andrew Salmon, 12Aug2011.)



Theorem  elintab 3750* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 30Aug1993.)



Theorem  elintrab 3751* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 17Oct1999.)



Theorem  elintrabg 3752* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 17Feb2007.)



Theorem  int0 3753 
The intersection of the empty set is the universal class. Exercise 2 of
[TakeutiZaring] p. 44.
(Contributed by NM, 18Aug1993.)



Theorem  intss1 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, 18Nov1995.)



Theorem  ssint 3755* 
Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
and its converse. (Contributed by NM, 14Oct1999.)



Theorem  ssintab 3756* 
Subclass of the intersection of a class abstraction. (Contributed by
NM, 31Jul2006.) (Proof shortened by Andrew Salmon, 9Jul2011.)



Theorem  ssintub 3757* 
Subclass of the least upper bound. (Contributed by NM, 8Aug2000.)



Theorem  ssmin 3758* 
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10Aug2006.)



Theorem  intmin 3759* 
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 3760 
Intersection of subclasses. (Contributed by NM, 14Oct1999.)



Theorem  intssunim 3761* 
The intersection of an inhabited set is a subclass of its union.
(Contributed by NM, 29Jul2006.)



Theorem  ssintrab 3762* 
Subclass of the intersection of a restricted class builder.
(Contributed by NM, 30Jan2015.)



Theorem  intssuni2m 3763* 
Subclass relationship for intersection and union. (Contributed by Jim
Kingdon, 14Aug2018.)



Theorem  intminss 3764* 
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 3765* 
Any set is the smallest of all sets that include it. (Contributed by
NM, 20Sep2003.)



Theorem  intmin3 3766* 
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 3767* 
Elimination of a conjunct in a class intersection. (Contributed by NM,
31Jul2006.)



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



Theorem  int0el 3769 
The intersection of a class containing the empty set is empty.
(Contributed by NM, 24Apr2004.)



Theorem  intun 3770 
The class intersection of the union of two classes. Theorem 78 of
[Suppes] p. 42. (Contributed by NM,
22Sep2002.)



Theorem  intpr 3771 
The intersection of a pair is the intersection of its members. Theorem
71 of [Suppes] p. 42. (Contributed by
NM, 14Oct1999.)



Theorem  intprg 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,
27Apr2008.)



Theorem  intsng 3773 
Intersection of a singleton. (Contributed by Stefan O'Rear,
22Feb2015.)



Theorem  intsn 3774 
The intersection of a singleton is its member. Theorem 70 of [Suppes]
p. 41. (Contributed by NM, 29Sep2002.)



Theorem  uniintsnr 3775* 
The union and intersection of a singleton are equal. See also eusn 3565.
(Contributed by Jim Kingdon, 14Aug2018.)



Theorem  uniintabim 3776 
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 3777 
Theorem joining a singleton to an intersection. (Contributed by NM,
29Sep2002.)



Theorem  rint0 3778 
Relative intersection of an empty set. (Contributed by Stefan O'Rear,
3Apr2015.)



Theorem  elrint 3779* 
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3Apr2015.)



Theorem  elrint2 3780* 
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3Apr2015.)



2.1.20 Indexed union and
intersection


Syntax  ciun 3781 
Extend class notation to include indexed union. Note: Historically
(prior to 21Oct2005), 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.



Syntax  ciin 3782 
Extend class notation to include indexed intersection. Note:
Historically (prior to 21Oct2005), 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.



Definition  dfiun 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, 27Jun1998.)



Definition  dfiin 3784* 
Define indexed intersection. Definition of [Stoll] p. 45. See the
remarks for its sibling operation of indexed union dfiun 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,
27Jun1998.)



Theorem  eliun 3785* 
Membership in indexed union. (Contributed by NM, 3Sep2003.)



Theorem  eliin 3786* 
Membership in indexed intersection. (Contributed by NM, 3Sep2003.)



Theorem  iuncom 3787* 
Commutation of indexed unions. (Contributed by NM, 18Dec2008.)



Theorem  iuncom4 3788 
Commutation of union with indexed union. (Contributed by Mario
Carneiro, 18Jan2014.)



Theorem  iunconstm 3789* 
Indexed union of a constant class, i.e. where does not depend on
. (Contributed
by Jim Kingdon, 15Aug2018.)



Theorem  iinconstm 3790* 
Indexed intersection of a constant class, i.e. where does not
depend on .
(Contributed by Jim Kingdon, 19Dec2018.)



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



Theorem  iunss1 3792* 
Subclass theorem for indexed union. (Contributed by NM, 10Dec2004.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iinss1 3793* 
Subclass theorem for indexed union. (Contributed by NM,
24Jan2012.)



Theorem  iuneq1 3794* 
Equality theorem for indexed union. (Contributed by NM,
27Jun1998.)



Theorem  iineq1 3795* 
Equality theorem for restricted existential quantifier. (Contributed by
NM, 27Jun1998.)



Theorem  ss2iun 3796 
Subclass theorem for indexed union. (Contributed by NM, 26Nov2003.)
(Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iuneq2 3797 
Equality theorem for indexed union. (Contributed by NM,
22Oct2003.)



Theorem  iineq2 3798 
Equality theorem for indexed intersection. (Contributed by NM,
22Oct2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)



Theorem  iuneq2i 3799 
Equality inference for indexed union. (Contributed by NM,
22Oct2003.)



Theorem  iineq2i 3800 
Equality inference for indexed intersection. (Contributed by NM,
22Oct2003.)

