Theorem List for Intuitionistic Logic Explorer - 3701-3800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | pwpw0ss 3701 |
Compute the power set of the power set of the empty set. (See pw0 3637
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.)
|
⊢ {∅, {∅}} ⊆ 𝒫
{∅} |
|
Theorem | pwprss 3702 |
The power set of an unordered pair. (Contributed by Jim Kingdon,
13-Aug-2018.)
|
⊢ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵} |
|
Theorem | pwtpss 3703 |
The power set of an unordered triple. (Contributed by Jim Kingdon,
13-Aug-2018.)
|
⊢ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ⊆ 𝒫 {𝐴, 𝐵, 𝐶} |
|
Theorem | pwpwpw0ss 3704 |
Compute the power set of the power set of the power set of the empty set.
(See also pw0 3637 and pwpw0ss 3701.) (Contributed by Jim Kingdon,
13-Aug-2018.)
|
⊢ ({∅, {∅}} ∪ {{{∅}},
{∅, {∅}}}) ⊆ 𝒫 {∅, {∅}} |
|
Theorem | pwv 3705 |
The power class of the universe is the universe. Exercise 4.12(d) of
[Mendelson] p. 235. (Contributed by NM,
14-Sep-2003.)
|
⊢ 𝒫 V = V |
|
2.1.18 The union of a class
|
|
Syntax | cuni 3706 |
Extend class notation to include the union of a class (read: 'union
𝐴')
|
class ∪ 𝐴 |
|
Definition | df-uni 3707* |
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 3045. (Contributed by NM, 23-Aug-1993.)
|
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} |
|
Theorem | dfuni2 3708* |
Alternate definition of class union. (Contributed by NM,
28-Jun-1998.)
|
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
|
Theorem | eluni 3709* |
Membership in class union. (Contributed by NM, 22-May-1994.)
|
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
|
Theorem | eluni2 3710* |
Membership in class union. Restricted quantifier version. (Contributed
by NM, 31-Aug-1999.)
|
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
|
Theorem | elunii 3711 |
Membership in class union. (Contributed by NM, 24-Mar-1995.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶) |
|
Theorem | nfuni 3712 |
Bound-variable hypothesis builder for union. (Contributed by NM,
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥∪
𝐴 |
|
Theorem | nfunid 3713 |
Deduction version of nfuni 3712. (Contributed by NM, 18-Feb-2013.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
|
Theorem | csbunig 3714 |
Distribute proper substitution through the union of a class.
(Contributed by Alan Sare, 10-Nov-2012.)
|
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪
𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵) |
|
Theorem | unieq 3715 |
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.)
|
⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪
𝐵) |
|
Theorem | unieqi 3716 |
Inference of equality of two class unions. (Contributed by NM,
30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∪
𝐴 = ∪ 𝐵 |
|
Theorem | unieqd 3717 |
Deduction of equality of two class unions. (Contributed by NM,
21-Apr-1995.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 = ∪
𝐵) |
|
Theorem | eluniab 3718* |
Membership in union of a class abstraction. (Contributed by NM,
11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
|
⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) |
|
Theorem | elunirab 3719* |
Membership in union of a class abstraction. (Contributed by NM,
4-Oct-2006.)
|
⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑)) |
|
Theorem | unipr 3720 |
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 23-Aug-1993.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ ∪
{𝐴, 𝐵} = (𝐴 ∪ 𝐵) |
|
Theorem | uniprg 3721 |
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 25-Aug-2006.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪
{𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
|
Theorem | unisn 3722 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 30-Aug-1993.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ ∪
{𝐴} = 𝐴 |
|
Theorem | unisng 3723 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 13-Aug-2002.)
|
⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) |
|
Theorem | dfnfc2 3724* |
An alternate statement of the effective freeness of a class 𝐴, when
it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
|
⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) |
|
Theorem | uniun 3725 |
The class union of the union of two classes. Theorem 8.3 of [Quine]
p. 53. (Contributed by NM, 20-Aug-1993.)
|
⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵) |
|
Theorem | uniin 3726 |
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.)
|
⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪
𝐴 ∩ ∪ 𝐵) |
|
Theorem | uniss 3727 |
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.)
|
⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | ssuni 3728 |
Subclass relationship for class union. (Contributed by NM,
24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶) |
|
Theorem | unissi 3729 |
Subclass relationship for subclass union. Inference form of uniss 3727.
(Contributed by David Moews, 1-May-2017.)
|
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ∪
𝐴 ⊆ ∪ 𝐵 |
|
Theorem | unissd 3730 |
Subclass relationship for subclass union. Deduction form of uniss 3727.
(Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | uni0b 3731 |
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.)
|
⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆
{∅}) |
|
Theorem | uni0c 3732* |
The union of a set is empty iff all of its members are empty.
(Contributed by NM, 16-Aug-2006.)
|
⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) |
|
Theorem | uni0 3733 |
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.)
|
⊢ ∪ ∅ =
∅ |
|
Theorem | elssuni 3734 |
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.)
|
⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | unissel 3735 |
Condition turning a subclass relationship for union into an equality.
(Contributed by NM, 18-Jul-2006.)
|
⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) |
|
Theorem | unissb 3736* |
Relationship involving membership, subset, and union. Exercise 5 of
[Enderton] p. 26 and its converse.
(Contributed by NM, 20-Sep-2003.)
|
⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
|
Theorem | uniss2 3737* |
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.)
|
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | unidif 3738* |
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,
22-Mar-2004.)
|
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴) |
|
Theorem | ssunieq 3739* |
Relationship implying union. (Contributed by NM, 10-Nov-1999.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵) |
|
Theorem | unimax 3740* |
Any member of a class is the largest of those members that it includes.
(Contributed by NM, 13-Aug-2002.)
|
⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴) |
|
2.1.19 The intersection of a class
|
|
Syntax | cint 3741 |
Extend class notation to include the intersection of a class (read:
'intersect 𝐴').
|
class ∩ 𝐴 |
|
Definition | df-int 3742* |
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 3047.
(Contributed by NM, 18-Aug-1993.)
|
⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} |
|
Theorem | dfint2 3743* |
Alternate definition of class intersection. (Contributed by NM,
28-Jun-1998.)
|
⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
|
Theorem | inteq 3744 |
Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
|
⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩
𝐵) |
|
Theorem | inteqi 3745 |
Equality inference for class intersection. (Contributed by NM,
2-Sep-2003.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∩
𝐴 = ∩ 𝐵 |
|
Theorem | inteqd 3746 |
Equality deduction for class intersection. (Contributed by NM,
2-Sep-2003.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∩ 𝐴 = ∩
𝐵) |
|
Theorem | elint 3747* |
Membership in class intersection. (Contributed by NM, 21-May-1994.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
|
Theorem | elint2 3748* |
Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
|
Theorem | elintg 3749* |
Membership in class intersection, with the sethood requirement expressed
as an antecedent. (Contributed by NM, 20-Nov-2003.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) |
|
Theorem | elinti 3750 |
Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
(Proof shortened by Andrew Salmon, 9-Jul-2011.)
|
⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
|
Theorem | nfint 3751 |
Bound-variable hypothesis builder for intersection. (Contributed by NM,
2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥∩
𝐴 |
|
Theorem | elintab 3752* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 30-Aug-1993.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
|
Theorem | elintrab 3753* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 17-Oct-1999.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)) |
|
Theorem | elintrabg 3754* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 17-Feb-2007.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))) |
|
Theorem | int0 3755 |
The intersection of the empty set is the universal class. Exercise 2 of
[TakeutiZaring] p. 44.
(Contributed by NM, 18-Aug-1993.)
|
⊢ ∩ ∅ =
V |
|
Theorem | intss1 3756 |
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.)
|
⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) |
|
Theorem | ssint 3757* |
Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
and its converse. (Contributed by NM, 14-Oct-1999.)
|
⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) |
|
Theorem | ssintab 3758* |
Subclass of the intersection of a class abstraction. (Contributed by
NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|
⊢ (𝐴 ⊆ ∩
{𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) |
|
Theorem | ssintub 3759* |
Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|
⊢ 𝐴 ⊆ ∩
{𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} |
|
Theorem | ssmin 3760* |
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10-Aug-2006.)
|
⊢ 𝐴 ⊆ ∩
{𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} |
|
Theorem | intmin 3761* |
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.)
|
⊢ (𝐴 ∈ 𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} = 𝐴) |
|
Theorem | intss 3762 |
Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|
⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
|
Theorem | intssunim 3763* |
The intersection of an inhabited set is a subclass of its union.
(Contributed by NM, 29-Jul-2006.)
|
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ ∪ 𝐴) |
|
Theorem | ssintrab 3764* |
Subclass of the intersection of a restricted class builder.
(Contributed by NM, 30-Jan-2015.)
|
⊢ (𝐴 ⊆ ∩
{𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ⊆ 𝑥)) |
|
Theorem | intssuni2m 3765* |
Subclass relationship for intersection and union. (Contributed by Jim
Kingdon, 14-Aug-2018.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | intminss 3766* |
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.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∩
{𝑥 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐴) |
|
Theorem | intmin2 3767* |
Any set is the smallest of all sets that include it. (Contributed by
NM, 20-Sep-2003.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ ∩
{𝑥 ∣ 𝐴 ⊆ 𝑥} = 𝐴 |
|
Theorem | intmin3 3768* |
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.)
|
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) |
|
Theorem | intmin4 3769* |
Elimination of a conjunct in a class intersection. (Contributed by NM,
31-Jul-2006.)
|
⊢ (𝐴 ⊆ ∩
{𝑥 ∣ 𝜑} → ∩
{𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} = ∩ {𝑥 ∣ 𝜑}) |
|
Theorem | intab 3770* |
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, 28-Jul-2006.)
(Proof shortened by
Mario Carneiro, 14-Nov-2016.)
|
⊢ 𝐴 ∈ V & ⊢ {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)} ∈ V ⇒ ⊢ ∩
{𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} = {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)} |
|
Theorem | int0el 3771 |
The intersection of a class containing the empty set is empty.
(Contributed by NM, 24-Apr-2004.)
|
⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
|
Theorem | intun 3772 |
The class intersection of the union of two classes. Theorem 78 of
[Suppes] p. 42. (Contributed by NM,
22-Sep-2002.)
|
⊢ ∩ (𝐴 ∪ 𝐵) = (∩ 𝐴 ∩ ∩ 𝐵) |
|
Theorem | intpr 3773 |
The intersection of a pair is the intersection of its members. Theorem
71 of [Suppes] p. 42. (Contributed by
NM, 14-Oct-1999.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ ∩
{𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
|
Theorem | intprg 3774 |
The intersection of a pair is the intersection of its members. Closed
form of intpr 3773. Theorem 71 of [Suppes] p. 42. (Contributed by FL,
27-Apr-2008.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩
{𝐴, 𝐵} = (𝐴 ∩ 𝐵)) |
|
Theorem | intsng 3775 |
Intersection of a singleton. (Contributed by Stefan O'Rear,
22-Feb-2015.)
|
⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
|
Theorem | intsn 3776 |
The intersection of a singleton is its member. Theorem 70 of [Suppes]
p. 41. (Contributed by NM, 29-Sep-2002.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ ∩
{𝐴} = 𝐴 |
|
Theorem | uniintsnr 3777* |
The union and intersection of a singleton are equal. See also eusn 3567.
(Contributed by Jim Kingdon, 14-Aug-2018.)
|
⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩
𝐴) |
|
Theorem | uniintabim 3778 |
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, 14-Aug-2018.)
|
⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) |
|
Theorem | intunsn 3779 |
Theorem joining a singleton to an intersection. (Contributed by NM,
29-Sep-2002.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ ∩
(𝐴 ∪ {𝐵}) = (∩ 𝐴
∩ 𝐵) |
|
Theorem | rint0 3780 |
Relative intersection of an empty set. (Contributed by Stefan O'Rear,
3-Apr-2015.)
|
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
|
Theorem | elrint 3781* |
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3-Apr-2015.)
|
⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) |
|
Theorem | elrint2 3782* |
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3-Apr-2015.)
|
⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) |
|
2.1.20 Indexed union and
intersection
|
|
Syntax | ciun 3783 |
Extend class notation to include indexed union. Note: Historically
(prior to 21-Oct-2005), set.mm used the notation ∪ 𝑥
∈ 𝐴𝐵, with
the same union symbol as cuni 3706. 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 3784 |
Extend class notation to include indexed intersection. Note:
Historically (prior to 21-Oct-2005), set.mm used the notation
∩ 𝑥 ∈ 𝐴𝐵, with the same intersection symbol
as cint 3741. 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 | df-iun 3785* |
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 3817. Theorem uniiun 3836 provides
a definition of ordinary union in terms of indexed union. (Contributed
by NM, 27-Jun-1998.)
|
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
|
Definition | df-iin 3786* |
Define indexed intersection. Definition of [Stoll] p. 45. See the
remarks for its sibling operation of indexed union df-iun 3785. An
alternate definition tying indexed intersection to ordinary intersection
is dfiin2 3818. Theorem intiin 3837 provides a definition of ordinary
intersection in terms of indexed intersection. (Contributed by NM,
27-Jun-1998.)
|
⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
|
Theorem | eliun 3787* |
Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
|
⊢ (𝐴 ∈ ∪
𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
|
Theorem | eliin 3788* |
Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩
𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
|
Theorem | iuncom 3789* |
Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
|
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪
𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 |
|
Theorem | iuncom4 3790 |
Commutation of union with indexed union. (Contributed by Mario
Carneiro, 18-Jan-2014.)
|
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪
∪ 𝑥 ∈ 𝐴 𝐵 |
|
Theorem | iunconstm 3791* |
Indexed union of a constant class, i.e. where 𝐵 does not depend on
𝑥. (Contributed by Jim Kingdon,
15-Aug-2018.)
|
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∪
𝑥 ∈ 𝐴 𝐵 = 𝐵) |
|
Theorem | iinconstm 3792* |
Indexed intersection of a constant class, i.e. where 𝐵 does not
depend on 𝑥. (Contributed by Jim Kingdon,
19-Dec-2018.)
|
⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩
𝑥 ∈ 𝐴 𝐵 = 𝐵) |
|
Theorem | iuniin 3793* |
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.)
|
⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 |
|
Theorem | iunss1 3794* |
Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.)
(Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ (𝐴 ⊆ 𝐵 → ∪
𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶) |
|
Theorem | iinss1 3795* |
Subclass theorem for indexed union. (Contributed by NM,
24-Jan-2012.)
|
⊢ (𝐴 ⊆ 𝐵 → ∩
𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iuneq1 3796* |
Equality theorem for indexed union. (Contributed by NM,
27-Jun-1998.)
|
⊢ (𝐴 = 𝐵 → ∪
𝑥 ∈ 𝐴 𝐶 = ∪
𝑥 ∈ 𝐵 𝐶) |
|
Theorem | iineq1 3797* |
Equality theorem for restricted existential quantifier. (Contributed by
NM, 27-Jun-1998.)
|
⊢ (𝐴 = 𝐵 → ∩
𝑥 ∈ 𝐴 𝐶 = ∩
𝑥 ∈ 𝐵 𝐶) |
|
Theorem | ss2iun 3798 |
Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.)
(Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪
𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iuneq2 3799 |
Equality theorem for indexed union. (Contributed by NM,
22-Oct-2003.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪
𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iineq2 3800 |
Equality theorem for indexed intersection. (Contributed by NM,
22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩
𝑥 ∈ 𝐴 𝐵 = ∩
𝑥 ∈ 𝐴 𝐶) |