Theorem List for Intuitionistic Logic Explorer - 3601-3700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | oteq2 3601 |
Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|
⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) |
|
Theorem | oteq3 3602 |
Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|
⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) |
|
Theorem | oteq1d 3603 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) |
|
Theorem | oteq2d 3604 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) |
|
Theorem | oteq3d 3605 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) |
|
Theorem | oteq123d 3606 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷)
& ⊢ (𝜑 → 𝐸 = 𝐹) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
|
Theorem | nfop 3607 |
Bound-variable hypothesis builder for ordered pairs. (Contributed by
NM, 14-Nov-1995.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 |
|
Theorem | nfopd 3608 |
Deduction version of bound-variable hypothesis builder nfop 3607.
This
shows how the deduction version of a not-free theorem such as nfop 3607
can
be created from the corresponding not-free inference theorem.
(Contributed by NM, 4-Feb-2008.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴)
& ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
|
Theorem | opid 3609 |
The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's
representation.
(Contributed by FL, 28-Dec-2011.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
|
Theorem | ralunsn 3610* |
Restricted quantification over the union of a set and a singleton, using
implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
(Revised by Mario Carneiro, 23-Apr-2015.)
|
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
|
Theorem | 2ralunsn 3611* |
Double restricted quantification over the union of a set and a
singleton, using implicit substitution. (Contributed by Paul Chapman,
17-Nov-2012.)
|
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))) |
|
Theorem | opprc 3612 |
Expansion of an ordered pair when either member is a proper class.
(Contributed by Mario Carneiro, 26-Apr-2015.)
|
⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) |
|
Theorem | opprc1 3613 |
Expansion of an ordered pair when the first member is a proper class. See
also opprc 3612. (Contributed by NM, 10-Apr-2004.) (Revised
by Mario
Carneiro, 26-Apr-2015.)
|
⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝐵〉 = ∅) |
|
Theorem | opprc2 3614 |
Expansion of an ordered pair when the second member is a proper class.
See also opprc 3612. (Contributed by NM, 15-Nov-1994.) (Revised
by Mario
Carneiro, 26-Apr-2015.)
|
⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) |
|
Theorem | oprcl 3615 |
If an ordered pair has an element, then its arguments are sets.
(Contributed by Mario Carneiro, 26-Apr-2015.)
|
⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
|
Theorem | pwsnss 3616 |
The power set of a singleton. (Contributed by Jim Kingdon,
12-Aug-2018.)
|
⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴} |
|
Theorem | pwpw0ss 3617 |
Compute the power set of the power set of the empty set. (See pw0 3553
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 3618 |
The power set of an unordered pair. (Contributed by Jim Kingdon,
13-Aug-2018.)
|
⊢ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵} |
|
Theorem | pwtpss 3619 |
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 3620 |
Compute the power set of the power set of the power set of the empty set.
(See also pw0 3553 and pwpw0ss 3617.) (Contributed by Jim Kingdon,
13-Aug-2018.)
|
⊢ ({∅, {∅}} ∪ {{{∅}},
{∅, {∅}}}) ⊆ 𝒫 {∅, {∅}} |
|
Theorem | pwv 3621 |
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 3622 |
Extend class notation to include the union of a class (read: 'union
𝐴')
|
class ∪ 𝐴 |
|
Definition | df-uni 3623* |
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 2987. (Contributed by NM, 23-Aug-1993.)
|
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} |
|
Theorem | dfuni2 3624* |
Alternate definition of class union. (Contributed by NM,
28-Jun-1998.)
|
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
|
Theorem | eluni 3625* |
Membership in class union. (Contributed by NM, 22-May-1994.)
|
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
|
Theorem | eluni2 3626* |
Membership in class union. Restricted quantifier version. (Contributed
by NM, 31-Aug-1999.)
|
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
|
Theorem | elunii 3627 |
Membership in class union. (Contributed by NM, 24-Mar-1995.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶) |
|
Theorem | nfuni 3628 |
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 3629 |
Deduction version of nfuni 3628. (Contributed by NM, 18-Feb-2013.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
|
Theorem | csbunig 3630 |
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 3631 |
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 3632 |
Inference of equality of two class unions. (Contributed by NM,
30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∪
𝐴 = ∪ 𝐵 |
|
Theorem | unieqd 3633 |
Deduction of equality of two class unions. (Contributed by NM,
21-Apr-1995.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 = ∪
𝐵) |
|
Theorem | eluniab 3634* |
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 3635* |
Membership in union of a class abstraction. (Contributed by NM,
4-Oct-2006.)
|
⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑)) |
|
Theorem | unipr 3636 |
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 3637 |
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 3638 |
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 3639 |
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 3640* |
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 3641 |
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 3642 |
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 3643 |
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 3644 |
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 3645 |
Subclass relationship for subclass union. Inference form of uniss 3643.
(Contributed by David Moews, 1-May-2017.)
|
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ∪
𝐴 ⊆ ∪ 𝐵 |
|
Theorem | unissd 3646 |
Subclass relationship for subclass union. Deduction form of uniss 3643.
(Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | uni0b 3647 |
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 3648* |
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 3649 |
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 3650 |
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 3651 |
Condition turning a subclass relationship for union into an equality.
(Contributed by NM, 18-Jul-2006.)
|
⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) |
|
Theorem | unissb 3652* |
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 3653* |
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 3654* |
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 3655* |
Relationship implying union. (Contributed by NM, 10-Nov-1999.)
|
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵) |
|
Theorem | unimax 3656* |
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 3657 |
Extend class notation to include the intersection of a class (read:
'intersect 𝐴').
|
class ∩ 𝐴 |
|
Definition | df-int 3658* |
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 2989.
(Contributed by NM, 18-Aug-1993.)
|
⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} |
|
Theorem | dfint2 3659* |
Alternate definition of class intersection. (Contributed by NM,
28-Jun-1998.)
|
⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
|
Theorem | inteq 3660 |
Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
|
⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩
𝐵) |
|
Theorem | inteqi 3661 |
Equality inference for class intersection. (Contributed by NM,
2-Sep-2003.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∩
𝐴 = ∩ 𝐵 |
|
Theorem | inteqd 3662 |
Equality deduction for class intersection. (Contributed by NM,
2-Sep-2003.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∩ 𝐴 = ∩
𝐵) |
|
Theorem | elint 3663* |
Membership in class intersection. (Contributed by NM, 21-May-1994.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) |
|
Theorem | elint2 3664* |
Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
|
Theorem | elintg 3665* |
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 3666 |
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 3667 |
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 3668* |
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 3669* |
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 3670* |
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 3671 |
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 3672 |
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 3673* |
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 3674* |
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 3675* |
Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|
⊢ 𝐴 ⊆ ∩
{𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} |
|
Theorem | ssmin 3676* |
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10-Aug-2006.)
|
⊢ 𝐴 ⊆ ∩
{𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} |
|
Theorem | intmin 3677* |
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 3678 |
Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|
⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
|
Theorem | intssunim 3679* |
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 3680* |
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 3681* |
Subclass relationship for intersection and union. (Contributed by Jim
Kingdon, 14-Aug-2018.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ ∪ 𝐵) |
|
Theorem | intminss 3682* |
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 3683* |
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 3684* |
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 3685* |
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 3686* |
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 3687 |
The intersection of a class containing the empty set is empty.
(Contributed by NM, 24-Apr-2004.)
|
⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
|
Theorem | intun 3688 |
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 3689 |
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 3690 |
The intersection of a pair is the intersection of its members. Closed
form of intpr 3689. Theorem 71 of [Suppes] p. 42. (Contributed by FL,
27-Apr-2008.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩
{𝐴, 𝐵} = (𝐴 ∩ 𝐵)) |
|
Theorem | intsng 3691 |
Intersection of a singleton. (Contributed by Stefan O'Rear,
22-Feb-2015.)
|
⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
|
Theorem | intsn 3692 |
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 3693* |
The union and intersection of a singleton are equal. See also eusn 3485.
(Contributed by Jim Kingdon, 14-Aug-2018.)
|
⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩
𝐴) |
|
Theorem | uniintabim 3694 |
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 3695 |
Theorem joining a singleton to an intersection. (Contributed by NM,
29-Sep-2002.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ ∩
(𝐴 ∪ {𝐵}) = (∩ 𝐴
∩ 𝐵) |
|
Theorem | rint0 3696 |
Relative intersection of an empty set. (Contributed by Stefan O'Rear,
3-Apr-2015.)
|
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
|
Theorem | elrint 3697* |
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3-Apr-2015.)
|
⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) |
|
Theorem | elrint2 3698* |
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 3699 |
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 3622. 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 3700 |
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 3657. 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 |