Theorem List for Intuitionistic Logic Explorer - 3801-3900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | elunii 3801 |
Membership in class union. (Contributed by NM, 24-Mar-1995.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶) |
|
Theorem | nfuni 3802 |
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 3803 |
Deduction version of nfuni 3802. (Contributed by NM, 18-Feb-2013.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
|
Theorem | csbunig 3804 |
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 3805 |
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 3806 |
Inference of equality of two class unions. (Contributed by NM,
30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∪
𝐴 = ∪ 𝐵 |
|
Theorem | unieqd 3807 |
Deduction of equality of two class unions. (Contributed by NM,
21-Apr-1995.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 = ∪
𝐵) |
|
Theorem | eluniab 3808* |
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 3809* |
Membership in union of a class abstraction. (Contributed by NM,
4-Oct-2006.)
|
⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑)) |
|
Theorem | unipr 3810 |
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 3811 |
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 3812 |
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 3813 |
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 3814* |
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 3815 |
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 3816 |
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 3817 |
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 3818 |
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 3819 |
Subclass relationship for subclass union. Inference form of uniss 3817.
(Contributed by David Moews, 1-May-2017.)
|
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ∪
𝐴 ⊆ ∪ 𝐵 |
|
Theorem | unissd 3820 |
Subclass relationship for subclass union. Deduction form of uniss 3817.
(Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | uni0b 3821 |
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 3822* |
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 3823 |
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 3824 |
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 3825 |
Condition turning a subclass relationship for union into an equality.
(Contributed by NM, 18-Jul-2006.)
|
⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) |
|
Theorem | unissb 3826* |
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 3827* |
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 3828* |
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 3829* |
Relationship implying union. (Contributed by NM, 10-Nov-1999.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵) |
|
Theorem | unimax 3830* |
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 3831 |
Extend class notation to include the intersection of a class. Read:
"intersection (of) 𝐴".
|
class ∩ 𝐴 |
|
Definition | df-int 3832* |
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 3127.
(Contributed by NM, 18-Aug-1993.)
|
⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} |
|
Theorem | dfint2 3833* |
Alternate definition of class intersection. (Contributed by NM,
28-Jun-1998.)
|
⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
|
Theorem | inteq 3834 |
Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
|
⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩
𝐵) |
|
Theorem | inteqi 3835 |
Equality inference for class intersection. (Contributed by NM,
2-Sep-2003.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∩
𝐴 = ∩ 𝐵 |
|
Theorem | inteqd 3836 |
Equality deduction for class intersection. (Contributed by NM,
2-Sep-2003.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∩ 𝐴 = ∩
𝐵) |
|
Theorem | elint 3837* |
Membership in class intersection. (Contributed by NM, 21-May-1994.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
|
Theorem | elint2 3838* |
Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
|
Theorem | elintg 3839* |
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 3840 |
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 3841 |
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 3842* |
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 3843* |
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 3844* |
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 3845 |
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 3846 |
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 3847* |
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 3848* |
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 3849* |
Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|
⊢ 𝐴 ⊆ ∩
{𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} |
|
Theorem | ssmin 3850* |
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10-Aug-2006.)
|
⊢ 𝐴 ⊆ ∩
{𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} |
|
Theorem | intmin 3851* |
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 3852 |
Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|
⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
|
Theorem | intssunim 3853* |
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 3854* |
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 3855* |
Subclass relationship for intersection and union. (Contributed by Jim
Kingdon, 14-Aug-2018.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | intminss 3856* |
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 3857* |
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 3858* |
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 3859* |
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 3860* |
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 3861 |
The intersection of a class containing the empty set is empty.
(Contributed by NM, 24-Apr-2004.)
|
⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
|
Theorem | intun 3862 |
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 3863 |
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 3864 |
The intersection of a pair is the intersection of its members. Closed
form of intpr 3863. Theorem 71 of [Suppes] p. 42. (Contributed by FL,
27-Apr-2008.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩
{𝐴, 𝐵} = (𝐴 ∩ 𝐵)) |
|
Theorem | intsng 3865 |
Intersection of a singleton. (Contributed by Stefan O'Rear,
22-Feb-2015.)
|
⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
|
Theorem | intsn 3866 |
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 3867* |
The union and intersection of a singleton are equal. See also eusn 3657.
(Contributed by Jim Kingdon, 14-Aug-2018.)
|
⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩
𝐴) |
|
Theorem | uniintabim 3868 |
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 3869 |
Theorem joining a singleton to an intersection. (Contributed by NM,
29-Sep-2002.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ ∩
(𝐴 ∪ {𝐵}) = (∩ 𝐴
∩ 𝐵) |
|
Theorem | rint0 3870 |
Relative intersection of an empty set. (Contributed by Stefan O'Rear,
3-Apr-2015.)
|
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
|
Theorem | elrint 3871* |
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3-Apr-2015.)
|
⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) |
|
Theorem | elrint2 3872* |
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 3873 |
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 3796. 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 3874 |
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 3831. 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 3875* |
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 3907. Theorem uniiun 3926 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 3876* |
Define indexed intersection. Definition of [Stoll] p. 45. See the
remarks for its sibling operation of indexed union df-iun 3875. An
alternate definition tying indexed intersection to ordinary intersection
is dfiin2 3908. Theorem intiin 3927 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 3877* |
Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
|
⊢ (𝐴 ∈ ∪
𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
|
Theorem | eliin 3878* |
Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩
𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
|
Theorem | iuncom 3879* |
Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
|
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪
𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 |
|
Theorem | iuncom4 3880 |
Commutation of union with indexed union. (Contributed by Mario
Carneiro, 18-Jan-2014.)
|
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪
∪ 𝑥 ∈ 𝐴 𝐵 |
|
Theorem | iunconstm 3881* |
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 3882* |
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 3883* |
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 3884* |
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 3885* |
Subclass theorem for indexed union. (Contributed by NM,
24-Jan-2012.)
|
⊢ (𝐴 ⊆ 𝐵 → ∩
𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iuneq1 3886* |
Equality theorem for indexed union. (Contributed by NM,
27-Jun-1998.)
|
⊢ (𝐴 = 𝐵 → ∪
𝑥 ∈ 𝐴 𝐶 = ∪
𝑥 ∈ 𝐵 𝐶) |
|
Theorem | iineq1 3887* |
Equality theorem for restricted existential quantifier. (Contributed by
NM, 27-Jun-1998.)
|
⊢ (𝐴 = 𝐵 → ∩
𝑥 ∈ 𝐴 𝐶 = ∩
𝑥 ∈ 𝐵 𝐶) |
|
Theorem | ss2iun 3888 |
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 3889 |
Equality theorem for indexed union. (Contributed by NM,
22-Oct-2003.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪
𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iineq2 3890 |
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) |
|
Theorem | iuneq2i 3891 |
Equality inference for indexed union. (Contributed by NM,
22-Oct-2003.)
|
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶 |
|
Theorem | iineq2i 3892 |
Equality inference for indexed intersection. (Contributed by NM,
22-Oct-2003.)
|
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩
𝑥 ∈ 𝐴 𝐶 |
|
Theorem | iineq2d 3893 |
Equality deduction for indexed intersection. (Contributed by NM,
7-Dec-2011.)
|
⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∩
𝑥 ∈ 𝐴 𝐵 = ∩
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iuneq2dv 3894* |
Equality deduction for indexed union. (Contributed by NM,
3-Aug-2004.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∪
𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iineq2dv 3895* |
Equality deduction for indexed intersection. (Contributed by NM,
3-Aug-2004.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∩
𝑥 ∈ 𝐴 𝐵 = ∩
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iuneq1d 3896* |
Equality theorem for indexed union, deduction version. (Contributed by
Drahflow, 22-Oct-2015.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∪
𝑥 ∈ 𝐴 𝐶 = ∪
𝑥 ∈ 𝐵 𝐶) |
|
Theorem | iuneq12d 3897* |
Equality deduction for indexed union, deduction version. (Contributed
by Drahflow, 22-Oct-2015.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∪
𝑥 ∈ 𝐴 𝐶 = ∪
𝑥 ∈ 𝐵 𝐷) |
|
Theorem | iuneq2d 3898* |
Equality deduction for indexed union. (Contributed by Drahflow,
22-Oct-2015.)
|
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∪
𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | nfiunxy 3899* |
Bound-variable hypothesis builder for indexed union. (Contributed by
Mario Carneiro, 25-Jan-2014.)
|
⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 |
|
Theorem | nfiinxy 3900* |
Bound-variable hypothesis builder for indexed intersection.
(Contributed by Mario Carneiro, 25-Jan-2014.)
|
⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |