HomeHome Intuitionistic Logic Explorer
Theorem List (p. 37 of 75)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssint 3601* 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 3602* 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 3603* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
A {x BAx}
 
Theoremssmin 3604* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
A {x ∣ (Ax φ)}
 
Theoremintmin 3605* 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 3606 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
(AB B A)
 
Theoremintssunim 3607* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
(x x A A A)
 
Theoremssintrab 3608* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
(A {x Bφ} ↔ x B (φAx))
 
Theoremintssuni2m 3609* Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
((AB x x A) → A B)
 
Theoremintminss 3610* 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 3611* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
A V        {xAx} = A
 
Theoremintmin3 3612* 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 3613* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
(A {xφ} → {x ∣ (Ax φ)} = {xφ})
 
Theoremintab 3614* 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 3615 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
(∅ A A = ∅)
 
Theoremintun 3616 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 3617 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 3618 The intersection of a pair is the intersection of its members. Closed form of intpr 3617. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
((A 𝑉 B 𝑊) → {A, B} = (AB))
 
Theoremintsng 3619 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(A 𝑉 {A} = A)
 
Theoremintsn 3620 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 3621* The union and intersection of a singleton are equal. See also eusn 3414. (Contributed by Jim Kingdon, 14-Aug-2018.)
(x A = {x} → A = A)
 
Theoremuniintabim 3622 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 3623 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
B V        (A ∪ {B}) = ( AB)
 
Theoremrint0 3624 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (A 𝑋) = A)
 
Theoremelrint 3625* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 (A B) ↔ (𝑋 A y B 𝑋 y))
 
Theoremelrint2 3626* 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 3627 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 3550. 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 3628 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 3585. 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 3629* 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 3661. Theorem uniiun 3680 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 3630* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3629. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3662. Theorem intiin 3681 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 3631* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
(A x B 𝐶x B A 𝐶)
 
Theoremeliin 3632* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
(A 𝑉 → (A x B 𝐶x B A 𝐶))
 
Theoremiuncom 3633* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
x A y B 𝐶 = y B x A 𝐶
 
Theoremiuncom4 3634 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
x A B = x A B
 
Theoremiunconstm 3635* 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 3636* 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 3637* 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 3638* 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 3639* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
(AB x B 𝐶 x A 𝐶)
 
Theoremiuneq1 3640* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
(A = B x A 𝐶 = x B 𝐶)
 
Theoremiineq1 3641* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
(A = B x A 𝐶 = x B 𝐶)
 
Theoremss2iun 3642 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 3643 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
(x A B = 𝐶 x A B = x A 𝐶)
 
Theoremiineq2 3644 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 3645 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
(x AB = 𝐶)        x A B = x A 𝐶
 
Theoremiineq2i 3646 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
(x AB = 𝐶)        x A B = x A 𝐶
 
Theoremiineq2d 3647 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 3648* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
((φ x A) → B = 𝐶)       (φ x A B = x A 𝐶)
 
Theoremiineq2dv 3649* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
((φ x A) → B = 𝐶)       (φ x A B = x A 𝐶)
 
Theoremiuneq1d 3650* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(φA = B)       (φ x A 𝐶 = x B 𝐶)
 
Theoremiuneq12d 3651* 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 3652* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
(φB = 𝐶)       (φ x A B = x A 𝐶)
 
Theoremnfiunxy 3653* Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
yA    &   yB       y x A B
 
Theoremnfiinxy 3654* Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
yA    &   yB       y x A B
 
Theoremnfiunya 3655* Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
yA    &   yB       y x A B
 
Theoremnfiinya 3656* Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
yA    &   yB       y x A B
 
Theoremnfiu1 3657 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
x x A B
 
Theoremnfii1 3658 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
x x A B
 
Theoremdfiun2g 3659* 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 3660* 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 3661* 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 3662* 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 3663* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
x A y B 𝐶 = {zx A y B z 𝐶}
 
Theoremcbviun 3664* 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 3665* 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 3666* 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 3667* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
(x = yB = 𝐶)        x A B = y A 𝐶
 
Theoremiunss 3668* 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 3669* 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 3670 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 3671* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
(x = 𝐶B = 𝐷)       (𝐶 A𝐷 x A B)
 
Theoremiunss2 3672* 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 3581. (Contributed by NM, 9-Dec-2004.)
(x A y B 𝐶𝐷 x A 𝐶 y B 𝐷)
 
Theoremiunab 3673* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
x A {yφ} = {yx A φ}
 
Theoremiunrab 3674* 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 3675* 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 3676 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 3677* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
(𝐶 x A Bx A 𝐶B)
 
Theoremiinss 3678* 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 3679 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
(x A x A BB)
 
Theoremuniiun 3680* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
A = x A x
 
Theoremintiin 3681* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
A = x A x
 
Theoremiunid 3682* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
x A {x} = A
 
Theoremiun0 3683 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
x A ∅ = ∅
 
Theorem0iun 3684 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
x A = ∅
 
Theorem0iin 3685 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
x A = V
 
Theoremviin 3686* Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
x V A = {yx y A}
 
Theoremiunn0m 3687* There is an inhabited class in an indexed collection B(x) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
(x A y y By y x A B)
 
Theoremiinab 3688* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
x A {yφ} = {yx A φ}
 
Theoremiinrabm 3689* Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
(x x A x A {y Bφ} = {y Bx A φ})
 
Theoremiunin2 3690* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3680 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
x A (B𝐶) = (B x A 𝐶)
 
Theoremiunin1 3691* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3680 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
x A (𝐶B) = ( x A 𝐶B)
 
Theoremiundif2ss 3692* Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
x A (B𝐶) ⊆ (B x A 𝐶)
 
Theorem2iunin 3693* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
x A y B (𝐶𝐷) = ( x A 𝐶 y B 𝐷)
 
Theoremiindif2m 3694* Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x x A x A (B𝐶) = (B x A 𝐶))
 
Theoremiinin2m 3695* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x x A x A (B𝐶) = (B x A 𝐶))
 
Theoremiinin1m 3696* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x x A x A (𝐶B) = ( x A 𝐶B))
 
Theoremelriin 3697* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
(B (A x 𝑋 𝑆) ↔ (B A x 𝑋 B 𝑆))
 
Theoremriin0 3698* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (A x 𝑋 𝑆) = A)
 
Theoremriinm 3699* Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
((x 𝑋 𝑆A x x 𝑋) → (A x 𝑋 𝑆) = x 𝑋 𝑆)
 
Theoremiinxsng 3700* A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(x = AB = 𝐶)       (A 𝑉 x {A}B = 𝐶)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7411
  Copyright terms: Public domain < Previous  Next >