Type  Label  Description 
Statement 

Theorem  pwv 3701 
The power class of the universe is the universe. Exercise 4.12(d) of
[Mendelson] p. 235. (Contributed by NM,
14Sep2003.)

⊢ 𝒫 V = V 

2.1.18 The union of a class


Syntax  cuni 3702 
Extend class notation to include the union of a class (read: 'union
𝐴')

class ∪ 𝐴 

Definition  dfuni 3703* 
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 3041. (Contributed by NM, 23Aug1993.)

⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} 

Theorem  dfuni2 3704* 
Alternate definition of class union. (Contributed by NM,
28Jun1998.)

⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} 

Theorem  eluni 3705* 
Membership in class union. (Contributed by NM, 22May1994.)

⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) 

Theorem  eluni2 3706* 
Membership in class union. Restricted quantifier version. (Contributed
by NM, 31Aug1999.)

⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) 

Theorem  elunii 3707 
Membership in class union. (Contributed by NM, 24Mar1995.)

⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶) 

Theorem  nfuni 3708 
Boundvariable hypothesis builder for union. (Contributed by NM,
30Dec1996.) (Proof shortened by Andrew Salmon, 27Aug2011.)

⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥∪
𝐴 

Theorem  nfunid 3709 
Deduction version of nfuni 3708. (Contributed by NM, 18Feb2013.)

⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) 

Theorem  csbunig 3710 
Distribute proper substitution through the union of a class.
(Contributed by Alan Sare, 10Nov2012.)

⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪
𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵) 

Theorem  unieq 3711 
Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.
(Contributed by NM, 10Aug1993.) (Proof shortened by Andrew Salmon,
29Jun2011.)

⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪
𝐵) 

Theorem  unieqi 3712 
Inference of equality of two class unions. (Contributed by NM,
30Aug1993.)

⊢ 𝐴 = 𝐵 ⇒ ⊢ ∪
𝐴 = ∪ 𝐵 

Theorem  unieqd 3713 
Deduction of equality of two class unions. (Contributed by NM,
21Apr1995.)

⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 = ∪
𝐵) 

Theorem  eluniab 3714* 
Membership in union of a class abstraction. (Contributed by NM,
11Aug1994.) (Revised by Mario Carneiro, 14Nov2016.)

⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) 

Theorem  elunirab 3715* 
Membership in union of a class abstraction. (Contributed by NM,
4Oct2006.)

⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑)) 

Theorem  unipr 3716 
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 23Aug1993.)

⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ ∪
{𝐴, 𝐵} = (𝐴 ∪ 𝐵) 

Theorem  uniprg 3717 
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 25Aug2006.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪
{𝐴, 𝐵} = (𝐴 ∪ 𝐵)) 

Theorem  unisn 3718 
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 30Aug1993.)

⊢ 𝐴 ∈ V ⇒ ⊢ ∪
{𝐴} = 𝐴 

Theorem  unisng 3719 
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 13Aug2002.)

⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) 

Theorem  dfnfc2 3720* 
An alternate statement of the effective freeness of a class 𝐴, when
it is a set. (Contributed by Mario Carneiro, 14Oct2016.)

⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) 

Theorem  uniun 3721 
The class union of the union of two classes. Theorem 8.3 of [Quine]
p. 53. (Contributed by NM, 20Aug1993.)

⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵) 

Theorem  uniin 3722 
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.)

⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪
𝐴 ∩ ∪ 𝐵) 

Theorem  uniss 3723 
Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
(Contributed by NM, 22Mar1998.) (Proof shortened by Andrew Salmon,
29Jun2011.)

⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) 

Theorem  ssuni 3724 
Subclass relationship for class union. (Contributed by NM,
24May1994.) (Proof shortened by Andrew Salmon, 29Jun2011.)

⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶) 

Theorem  unissi 3725 
Subclass relationship for subclass union. Inference form of uniss 3723.
(Contributed by David Moews, 1May2017.)

⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ∪
𝐴 ⊆ ∪ 𝐵 

Theorem  unissd 3726 
Subclass relationship for subclass union. Deduction form of uniss 3723.
(Contributed by David Moews, 1May2017.)

⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵) 

Theorem  uni0b 3727 
The union of a set is empty iff the set is included in the singleton of
the empty set. (Contributed by NM, 12Sep2004.)

⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆
{∅}) 

Theorem  uni0c 3728* 
The union of a set is empty iff all of its members are empty.
(Contributed by NM, 16Aug2006.)

⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) 

Theorem  uni0 3729 
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 3730 
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.)

⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) 

Theorem  unissel 3731 
Condition turning a subclass relationship for union into an equality.
(Contributed by NM, 18Jul2006.)

⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) 

Theorem  unissb 3732* 
Relationship involving membership, subset, and union. Exercise 5 of
[Enderton] p. 26 and its converse.
(Contributed by NM, 20Sep2003.)

⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) 

Theorem  uniss2 3733* 
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.)

⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵) 

Theorem  unidif 3734* 
If the difference 𝐴 ∖ 𝐵 contains the largest members of
𝐴,
then
the union of the difference is the union of 𝐴. (Contributed by NM,
22Mar2004.)

⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴) 

Theorem  ssunieq 3735* 
Relationship implying union. (Contributed by NM, 10Nov1999.)

⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵) 

Theorem  unimax 3736* 
Any member of a class is the largest of those members that it includes.
(Contributed by NM, 13Aug2002.)

⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴) 

2.1.19 The intersection of a class


Syntax  cint 3737 
Extend class notation to include the intersection of a class (read:
'intersect 𝐴').

class ∩ 𝐴 

Definition  dfint 3738* 
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 3043.
(Contributed by NM, 18Aug1993.)

⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} 

Theorem  dfint2 3739* 
Alternate definition of class intersection. (Contributed by NM,
28Jun1998.)

⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} 

Theorem  inteq 3740 
Equality law for intersection. (Contributed by NM, 13Sep1999.)

⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩
𝐵) 

Theorem  inteqi 3741 
Equality inference for class intersection. (Contributed by NM,
2Sep2003.)

⊢ 𝐴 = 𝐵 ⇒ ⊢ ∩
𝐴 = ∩ 𝐵 

Theorem  inteqd 3742 
Equality deduction for class intersection. (Contributed by NM,
2Sep2003.)

⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∩ 𝐴 = ∩
𝐵) 

Theorem  elint 3743* 
Membership in class intersection. (Contributed by NM, 21May1994.)

⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) 

Theorem  elint2 3744* 
Membership in class intersection. (Contributed by NM, 14Oct1999.)

⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) 

Theorem  elintg 3745* 
Membership in class intersection, with the sethood requirement expressed
as an antecedent. (Contributed by NM, 20Nov2003.)

⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) 

Theorem  elinti 3746 
Membership in class intersection. (Contributed by NM, 14Oct1999.)
(Proof shortened by Andrew Salmon, 9Jul2011.)

⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) 

Theorem  nfint 3747 
Boundvariable hypothesis builder for intersection. (Contributed by NM,
2Feb1997.) (Proof shortened by Andrew Salmon, 12Aug2011.)

⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥∩
𝐴 

Theorem  elintab 3748* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 30Aug1993.)

⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) 

Theorem  elintrab 3749* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 17Oct1999.)

⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)) 

Theorem  elintrabg 3750* 
Membership in the intersection of a class abstraction. (Contributed by
NM, 17Feb2007.)

⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))) 

Theorem  int0 3751 
The intersection of the empty set is the universal class. Exercise 2 of
[TakeutiZaring] p. 44.
(Contributed by NM, 18Aug1993.)

⊢ ∩ ∅ =
V 

Theorem  intss1 3752 
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.)

⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) 

Theorem  ssint 3753* 
Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
and its converse. (Contributed by NM, 14Oct1999.)

⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) 

Theorem  ssintab 3754* 
Subclass of the intersection of a class abstraction. (Contributed by
NM, 31Jul2006.) (Proof shortened by Andrew Salmon, 9Jul2011.)

⊢ (𝐴 ⊆ ∩
{𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) 

Theorem  ssintub 3755* 
Subclass of the least upper bound. (Contributed by NM, 8Aug2000.)

⊢ 𝐴 ⊆ ∩
{𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} 

Theorem  ssmin 3756* 
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10Aug2006.)

⊢ 𝐴 ⊆ ∩
{𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} 

Theorem  intmin 3757* 
Any member of a class is the smallest of those members that include it.
(Contributed by NM, 13Aug2002.) (Proof shortened by Andrew Salmon,
9Jul2011.)

⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} = 𝐴) 

Theorem  intss 3758 
Intersection of subclasses. (Contributed by NM, 14Oct1999.)

⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) 

Theorem  intssunim 3759* 
The intersection of an inhabited set is a subclass of its union.
(Contributed by NM, 29Jul2006.)

⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ ∪ 𝐴) 

Theorem  ssintrab 3760* 
Subclass of the intersection of a restricted class builder.
(Contributed by NM, 30Jan2015.)

⊢ (𝐴 ⊆ ∩
{𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥)) 

Theorem  intssuni2m 3761* 
Subclass relationship for intersection and union. (Contributed by Jim
Kingdon, 14Aug2018.)

⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ 𝐵) 

Theorem  intminss 3762* 
Under subset ordering, the intersection of a restricted class
abstraction is less than or equal to any of its members. (Contributed
by NM, 7Sep2013.)

⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩
{𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴) 

Theorem  intmin2 3763* 
Any set is the smallest of all sets that include it. (Contributed by
NM, 20Sep2003.)

⊢ 𝐴 ∈ V ⇒ ⊢ ∩
{𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 

Theorem  intmin3 3764* 
Under subset ordering, the intersection of a class abstraction is less
than or equal to any of its members. (Contributed by NM,
3Jul2005.)

⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) 

Theorem  intmin4 3765* 
Elimination of a conjunct in a class intersection. (Contributed by NM,
31Jul2006.)

⊢ (𝐴 ⊆ ∩
{𝑥 ∣ 𝜑} → ∩
{𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑}) 

Theorem  intab 3766* 
The intersection of a special case of a class abstraction. 𝑦 may be
free in 𝜑 and 𝐴, which can be thought of
a 𝜑(𝑦) and
𝐴(𝑦). (Contributed by NM, 28Jul2006.)
(Proof shortened by
Mario Carneiro, 14Nov2016.)

⊢ 𝐴 ∈ V & ⊢ {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)} ∈ V ⇒ ⊢ ∩
{𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} = {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)} 

Theorem  int0el 3767 
The intersection of a class containing the empty set is empty.
(Contributed by NM, 24Apr2004.)

⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) 

Theorem  intun 3768 
The class intersection of the union of two classes. Theorem 78 of
[Suppes] p. 42. (Contributed by NM,
22Sep2002.)

⊢ ∩ (𝐴 ∪ 𝐵) = (∩ 𝐴 ∩ ∩ 𝐵) 

Theorem  intpr 3769 
The intersection of a pair is the intersection of its members. Theorem
71 of [Suppes] p. 42. (Contributed by
NM, 14Oct1999.)

⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ ∩
{𝐴, 𝐵} = (𝐴 ∩ 𝐵) 

Theorem  intprg 3770 
The intersection of a pair is the intersection of its members. Closed
form of intpr 3769. Theorem 71 of [Suppes] p. 42. (Contributed by FL,
27Apr2008.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩
{𝐴, 𝐵} = (𝐴 ∩ 𝐵)) 

Theorem  intsng 3771 
Intersection of a singleton. (Contributed by Stefan O'Rear,
22Feb2015.)

⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) 

Theorem  intsn 3772 
The intersection of a singleton is its member. Theorem 70 of [Suppes]
p. 41. (Contributed by NM, 29Sep2002.)

⊢ 𝐴 ∈ V ⇒ ⊢ ∩
{𝐴} = 𝐴 

Theorem  uniintsnr 3773* 
The union and intersection of a singleton are equal. See also eusn 3563.
(Contributed by Jim Kingdon, 14Aug2018.)

⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩
𝐴) 

Theorem  uniintabim 3774 
The union and the intersection of a class abstraction are equal if there
is a unique satisfying value of 𝜑(𝑥). (Contributed by Jim
Kingdon, 14Aug2018.)

⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) 

Theorem  intunsn 3775 
Theorem joining a singleton to an intersection. (Contributed by NM,
29Sep2002.)

⊢ 𝐵 ∈ V ⇒ ⊢ ∩
(𝐴 ∪ {𝐵}) = (∩ 𝐴
∩ 𝐵) 

Theorem  rint0 3776 
Relative intersection of an empty set. (Contributed by Stefan O'Rear,
3Apr2015.)

⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) 

Theorem  elrint 3777* 
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3Apr2015.)

⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) 

Theorem  elrint2 3778* 
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3Apr2015.)

⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) 

2.1.20 Indexed union and
intersection


Syntax  ciun 3779 
Extend class notation to include indexed union. Note: Historically
(prior to 21Oct2005), set.mm used the notation ∪ 𝑥
∈ 𝐴𝐵, with
the same union symbol as cuni 3702. 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.

class ∪ 𝑥 ∈ 𝐴 𝐵 

Syntax  ciin 3780 
Extend class notation to include indexed intersection. Note:
Historically (prior to 21Oct2005), set.mm used the notation
∩ 𝑥 ∈ 𝐴𝐵, with the same intersection symbol
as cint 3737. 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.

class ∩ 𝑥 ∈ 𝐴 𝐵 

Definition  dfiun 3781* 
Define indexed union. Definition indexed union in [Stoll] p. 45. In
most applications, 𝐴 is independent of 𝑥
(although this is not
required by the definition), and 𝐵 depends on 𝑥 i.e. can be read
informally as 𝐵(𝑥). We call 𝑥 the index, 𝐴 the
index
set, and 𝐵 the indexed set. In most books,
𝑥 ∈
𝐴 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 𝑥 and 𝐴 are in
the
same disjoint variable group (meaning 𝐴 cannot depend on 𝑥) and
that 𝐵 and 𝑥 do not share a disjoint
variable group (meaning
that can be thought of as 𝐵(𝑥) i.e. can be substituted with a
class expression containing 𝑥). An alternate definition tying
indexed union to ordinary union is dfiun2 3813. Theorem uniiun 3832 provides
a definition of ordinary union in terms of indexed union. (Contributed
by NM, 27Jun1998.)

⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} 

Definition  dfiin 3782* 
Define indexed intersection. Definition of [Stoll] p. 45. See the
remarks for its sibling operation of indexed union dfiun 3781. An
alternate definition tying indexed intersection to ordinary intersection
is dfiin2 3814. Theorem intiin 3833 provides a definition of ordinary
intersection in terms of indexed intersection. (Contributed by NM,
27Jun1998.)

⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} 

Theorem  eliun 3783* 
Membership in indexed union. (Contributed by NM, 3Sep2003.)

⊢ (𝐴 ∈ ∪
𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) 

Theorem  eliin 3784* 
Membership in indexed intersection. (Contributed by NM, 3Sep2003.)

⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩
𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) 

Theorem  iuncom 3785* 
Commutation of indexed unions. (Contributed by NM, 18Dec2008.)

⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪
𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 

Theorem  iuncom4 3786 
Commutation of union with indexed union. (Contributed by Mario
Carneiro, 18Jan2014.)

⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪
∪ 𝑥 ∈ 𝐴 𝐵 

Theorem  iunconstm 3787* 
Indexed union of a constant class, i.e. where 𝐵 does not depend on
𝑥. (Contributed by Jim Kingdon,
15Aug2018.)

⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∪
𝑥 ∈ 𝐴 𝐵 = 𝐵) 

Theorem  iinconstm 3788* 
Indexed intersection of a constant class, i.e. where 𝐵 does not
depend on 𝑥. (Contributed by Jim Kingdon,
19Dec2018.)

⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩
𝑥 ∈ 𝐴 𝐵 = 𝐵) 

Theorem  iuniin 3789* 
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.)

⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 

Theorem  iunss1 3790* 
Subclass theorem for indexed union. (Contributed by NM, 10Dec2004.)
(Proof shortened by Andrew Salmon, 25Jul2011.)

⊢ (𝐴 ⊆ 𝐵 → ∪
𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶) 

Theorem  iinss1 3791* 
Subclass theorem for indexed union. (Contributed by NM,
24Jan2012.)

⊢ (𝐴 ⊆ 𝐵 → ∩
𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) 

Theorem  iuneq1 3792* 
Equality theorem for indexed union. (Contributed by NM,
27Jun1998.)

⊢ (𝐴 = 𝐵 → ∪
𝑥 ∈ 𝐴 𝐶 = ∪
𝑥 ∈ 𝐵 𝐶) 

Theorem  iineq1 3793* 
Equality theorem for restricted existential quantifier. (Contributed by
NM, 27Jun1998.)

⊢ (𝐴 = 𝐵 → ∩
𝑥 ∈ 𝐴 𝐶 = ∩
𝑥 ∈ 𝐵 𝐶) 

Theorem  ss2iun 3794 
Subclass theorem for indexed union. (Contributed by NM, 26Nov2003.)
(Proof shortened by Andrew Salmon, 25Jul2011.)

⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪
𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) 

Theorem  iuneq2 3795 
Equality theorem for indexed union. (Contributed by NM,
22Oct2003.)

⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪
𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶) 

Theorem  iineq2 3796 
Equality theorem for indexed intersection. (Contributed by NM,
22Oct2003.) (Proof shortened by Andrew Salmon, 25Jul2011.)

⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩
𝑥 ∈ 𝐴 𝐵 = ∩
𝑥 ∈ 𝐴 𝐶) 

Theorem  iuneq2i 3797 
Equality inference for indexed union. (Contributed by NM,
22Oct2003.)

⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶 

Theorem  iineq2i 3798 
Equality inference for indexed intersection. (Contributed by NM,
22Oct2003.)

⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩
𝑥 ∈ 𝐴 𝐶 

Theorem  iineq2d 3799 
Equality deduction for indexed intersection. (Contributed by NM,
7Dec2011.)

⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∩
𝑥 ∈ 𝐴 𝐵 = ∩
𝑥 ∈ 𝐴 𝐶) 

Theorem  iuneq2dv 3800* 
Equality deduction for indexed union. (Contributed by NM,
3Aug2004.)

⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∪
𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶) 