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Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremunissb 3601* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
( ABx A xB)

Theoremuniss2 3602* 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 xy A B)

Theoremunidif 3603* If the difference AB 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 (AB)xy (AB) = A)

Theoremssunieq 3604* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
((A B x B xA) → A = B)

Theoremunimax 3605* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
(A B {x BxA} = A)

2.1.19  The intersection of a class

Syntaxcint 3606 Extend class notation to include the intersection of a class (read: 'intersect A').
class A

Definitiondf-int 3607* 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 2918. (Contributed by NM, 18-Aug-1993.)
A = {xy(y Ax y)}

Theoremdfint2 3608* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
A = {xy A x y}

Theoreminteq 3609 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
(A = B A = B)

Theoreminteqi 3610 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
A = B        A = B

Theoreminteqd 3611 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
(φA = B)       (φ A = B)

Theoremelint 3612* Membership in class intersection. (Contributed by NM, 21-May-1994.)
A V       (A Bx(x BA x))

Theoremelint2 3613* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
A V       (A Bx B A x)

Theoremelintg 3614* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
(A 𝑉 → (A Bx B A x))

Theoremelinti 3615 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(A B → (𝐶 BA 𝐶))

Theoremnfint 3616 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
xA       x A

Theoremelintab 3617* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
A V       (A {xφ} ↔ x(φA x))

Theoremelintrab 3618* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
A V       (A {x Bφ} ↔ x B (φA x))

Theoremelintrabg 3619* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
(A 𝑉 → (A {x Bφ} ↔ x B (φA x)))

Theoremint0 3620 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
∅ = V

Theoremintss1 3621 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 BA)

Theoremssint 3622* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
(A Bx B Ax)

Theoremssintab 3623* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(A {xφ} ↔ x(φAx))

Theoremssintub 3624* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
A {x BAx}

Theoremssmin 3625* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
A {x ∣ (Ax φ)}

Theoremintmin 3626* 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 BAx} = A)

Theoremintss 3627 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
(AB B A)

Theoremintssunim 3628* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
(x x A A A)

Theoremssintrab 3629* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
(A {x Bφ} ↔ x B (φAx))

Theoremintssuni2m 3630* Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
((AB x x A) → A B)

Theoremintminss 3631* 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 → (φψ))       ((A B ψ) → {x Bφ} ⊆ A)

Theoremintmin2 3632* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
A V        {xAx} = A

Theoremintmin3 3633* 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 → (φψ))    &   ψ       (A 𝑉 {xφ} ⊆ A)

Theoremintmin4 3634* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
(A {xφ} → {x ∣ (Ax φ)} = {xφ})

Theoremintab 3635* 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    &   {xy(φ x = A)} V        {xy(φA x)} = {xy(φ x = A)}

Theoremint0el 3636 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
(∅ A A = ∅)

Theoremintun 3637 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
(AB) = ( A B)

Theoremintpr 3638 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} = (AB)

Theoremintprg 3639 The intersection of a pair is the intersection of its members. Closed form of intpr 3638. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
((A 𝑉 B 𝑊) → {A, B} = (AB))

Theoremintsng 3640 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(A 𝑉 {A} = A)

Theoremintsn 3641 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
A V        {A} = A

Theoremuniintsnr 3642* The union and intersection of a singleton are equal. See also eusn 3435. (Contributed by Jim Kingdon, 14-Aug-2018.)
(x A = {x} → A = A)

Theoremuniintabim 3643 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 {xφ} = {xφ})

Theoremintunsn 3644 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
B V        (A ∪ {B}) = ( AB)

Theoremrint0 3645 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (A 𝑋) = A)

Theoremelrint 3646* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 (A B) ↔ (𝑋 A y B 𝑋 y))

Theoremelrint2 3647* 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

Syntaxciun 3648 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 3571. 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

Syntaxciin 3649 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 3606. 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

Definitiondf-iun 3650* 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 distinct variable group (meaning A cannot depend on x) and that B and x do not share a distinct 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 3682. Theorem uniiun 3701 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)
x A B = {yx A y B}

Definitiondf-iin 3651* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3650. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3683. Theorem intiin 3702 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
x A B = {yx A y B}

Theoremeliun 3652* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
(A x B 𝐶x B A 𝐶)

Theoremeliin 3653* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
(A 𝑉 → (A x B 𝐶x B A 𝐶))

Theoremiuncom 3654* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
x A y B 𝐶 = y B x A 𝐶

Theoremiuncom4 3655 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
x A B = x A B

Theoremiunconstm 3656* 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)

Theoremiinconstm 3657* 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)

Theoremiuniin 3658* 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

Theoremiunss1 3659* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(AB x A 𝐶 x B 𝐶)

Theoremiinss1 3660* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
(AB x B 𝐶 x A 𝐶)

Theoremiuneq1 3661* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
(A = B x A 𝐶 = x B 𝐶)

Theoremiineq1 3662* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
(A = B x A 𝐶 = x B 𝐶)

Theoremss2iun 3663 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(x A B𝐶 x A B x A 𝐶)

Theoremiuneq2 3664 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
(x A B = 𝐶 x A B = x A 𝐶)

Theoremiineq2 3665 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)

Theoremiuneq2i 3666 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
(x AB = 𝐶)        x A B = x A 𝐶

Theoremiineq2i 3667 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
(x AB = 𝐶)        x A B = x A 𝐶

Theoremiineq2d 3668 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
xφ    &   ((φ x A) → B = 𝐶)       (φ x A B = x A 𝐶)

Theoremiuneq2dv 3669* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
((φ x A) → B = 𝐶)       (φ x A B = x A 𝐶)

Theoremiineq2dv 3670* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
((φ x A) → B = 𝐶)       (φ x A B = x A 𝐶)

Theoremiuneq1d 3671* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(φA = B)       (φ x A 𝐶 = x B 𝐶)

Theoremiuneq12d 3672* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ x A 𝐶 = x B 𝐷)

Theoremiuneq2d 3673* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
(φB = 𝐶)       (φ x A B = x A 𝐶)

Theoremnfiunxy 3674* Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
yA    &   yB       y x A B

Theoremnfiinxy 3675* Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
yA    &   yB       y x A B

Theoremnfiunya 3676* Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
yA    &   yB       y x A B

Theoremnfiinya 3677* Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
yA    &   yB       y x A B

Theoremnfiu1 3678 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
x x A B

Theoremnfii1 3679 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
x x A B

Theoremdfiun2g 3680* Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(x A B 𝐶 x A B = {yx A y = B})

Theoremdfiin2g 3681* Alternate definition of indexed intersection when B is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
(x A B 𝐶 x A B = {yx A y = B})

Theoremdfiun2 3682* Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
B V        x A B = {yx A y = B}

Theoremdfiin2 3683* Alternate definition of indexed intersection when B is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
B V        x A B = {yx A y = B}

Theoremdfiunv2 3684* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
x A y B 𝐶 = {zx A y B z 𝐶}

Theoremcbviun 3685* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
yB    &   x𝐶    &   (x = yB = 𝐶)        x A B = y A 𝐶

Theoremcbviin 3686* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
yB    &   x𝐶    &   (x = yB = 𝐶)        x A B = y A 𝐶

Theoremcbviunv 3687* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)
(x = yB = 𝐶)        x A B = y A 𝐶

Theoremcbviinv 3688* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
(x = yB = 𝐶)        x A B = y A 𝐶

Theoremiunss 3689* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
( x A B𝐶x A B𝐶)

Theoremssiun 3690* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(x A 𝐶B𝐶 x A B)

Theoremssiun2 3691 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(x AB x A B)

Theoremssiun2s 3692* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
(x = 𝐶B = 𝐷)       (𝐶 A𝐷 x A B)

Theoremiunss2 3693* A subclass condition on the members of two indexed classes 𝐶(x) and 𝐷(y) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3602. (Contributed by NM, 9-Dec-2004.)
(x A y B 𝐶𝐷 x A 𝐶 y B 𝐷)

Theoremiunab 3694* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
x A {yφ} = {yx A φ}

Theoremiunrab 3695* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
x A {y Bφ} = {y Bx A φ}

Theoremiunxdif2 3696* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
(x = y𝐶 = 𝐷)       (x A y (AB)𝐶𝐷 y (AB)𝐷 = x A 𝐶)

Theoremssiinf 3697 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
x𝐶       (𝐶 x A Bx A 𝐶B)

Theoremssiin 3698* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
(𝐶 x A Bx A 𝐶B)

Theoremiinss 3699* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(x A B𝐶 x A B𝐶)

Theoremiinss2 3700 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
(x A x A BB)

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