Home Intuitionistic Logic ExplorerTheorem List (p. 38 of 86) < 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 - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiunxun 3701 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
x (AB)𝐶 = ( x A 𝐶 x B 𝐶)

Theoremiunxiun 3702* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
x y A B𝐶 = y A x B 𝐶

Theoremiinuniss 3703* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
(A B) ⊆ x B (Ax)

Theoremiununir 3704* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)
((A B) = x B (Ax) → (B = ∅ → A = ∅))

Theoremsspwuni 3705 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(A ⊆ 𝒫 B AB)

Theorempwssb 3706* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
(A ⊆ 𝒫 Bx A xB)

Theoremelpwuni 3707 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(B A → (A ⊆ 𝒫 B A = B))

Theoremiinpw 3708* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
𝒫 A = x A 𝒫 x

Theoremiunpwss 3709* Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
x A 𝒫 x ⊆ 𝒫 A

Theoremrintm 3710* Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝑋 ⊆ 𝒫 A x x 𝑋) → (A 𝑋) = 𝑋)

2.1.21  Disjointness

Syntaxwdisj 3711 Extend wff notation to include the statement that a family of classes B(x), for x A, is a disjoint family.
wff Disj x A B

Definitiondf-disj 3712* A collection of classes B(x) is disjoint when for each element y, it is in B(x) for at most one x. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
(Disj x A By∃*x A y B)

Theoremdfdisj2 3713* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
(Disj x A By∃*x(x A y B))

Theoremdisjss2 3714 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(x A B𝐶 → (Disj x A 𝐶Disj x A B))

Theoremdisjeq2 3715 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(x A B = 𝐶 → (Disj x A BDisj x A 𝐶))

Theoremdisjeq2dv 3716* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
((φ x A) → B = 𝐶)       (φ → (Disj x A BDisj x A 𝐶))

Theoremdisjss1 3717* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(AB → (Disj x B 𝐶Disj x A 𝐶))

Theoremdisjeq1 3718* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(A = B → (Disj x A 𝐶Disj x B 𝐶))

Theoremdisjeq1d 3719* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(φA = B)       (φ → (Disj x A 𝐶Disj x B 𝐶))

Theoremdisjeq12d 3720* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (Disj x A 𝐶Disj x B 𝐷))

Theoremcbvdisj 3721* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
yB    &   x𝐶    &   (x = yB = 𝐶)       (Disj x A BDisj y A 𝐶)

Theoremcbvdisjv 3722* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
(x = yB = 𝐶)       (Disj x A BDisj y A 𝐶)

Theoremnfdisjv 3723* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
yA    &   yB       yDisj x A B

Theoremnfdisj1 3724 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
xDisj x A B

Theoreminvdisj 3725* If there is a function 𝐶(y) such that 𝐶(y) = x for all y B(x), then the sets B(x) for distinct x A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
(x A y B 𝐶 = xDisj x A B)

Theoremsndisj 3726 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x A {x}

Theorem0disj 3727 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x A

Theoremdisjxsn 3728* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x {A}B

Theoremdisjx0 3729 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x B

2.1.22  Binary relations

Syntaxwbr 3730 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
wff A𝑅B

Definitiondf-br 3731 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes A and/or B are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when 𝑅 is a proper class (see for example iprc 4518). (Contributed by NM, 31-Dec-1993.)
(A𝑅B ↔ ⟨A, B 𝑅)

Theorembreq 3732 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (A𝑅BA𝑆B))

Theorembreq1 3733 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(A = B → (A𝑅𝐶B𝑅𝐶))

Theorembreq2 3734 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(A = B → (𝐶𝑅A𝐶𝑅B))

Theorembreq12 3735 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((A = B 𝐶 = 𝐷) → (A𝑅𝐶B𝑅𝐷))

Theorembreqi 3736 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (A𝑅BA𝑆B)

Theorembreq1i 3737 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
A = B       (A𝑅𝐶B𝑅𝐶)

Theorembreq2i 3738 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
A = B       (𝐶𝑅A𝐶𝑅B)

Theorembreq12i 3739 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
A = B    &   𝐶 = 𝐷       (A𝑅𝐶B𝑅𝐷)

Theorembreq1d 3740 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)       (φ → (A𝑅𝐶B𝑅𝐶))

Theorembreqd 3741 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(φA = B)       (φ → (𝐶A𝐷𝐶B𝐷))

Theorembreq2d 3742 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)       (φ → (𝐶𝑅A𝐶𝑅B))

Theorembreq12d 3743 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝑅𝐶B𝑅𝐷))

Theorembreq123d 3744 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(φA = B)    &   (φ𝑅 = 𝑆)    &   (φ𝐶 = 𝐷)       (φ → (A𝑅𝐶B𝑆𝐷))

Theorembreqan12d 3745 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((φ ψ) → (A𝑅𝐶B𝑅𝐷))

Theorembreqan12rd 3746 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((ψ φ) → (A𝑅𝐶B𝑅𝐷))

Theoremnbrne1 3747 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((A𝑅B ¬ A𝑅𝐶) → B𝐶)

Theoremnbrne2 3748 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((A𝑅𝐶 ¬ B𝑅𝐶) → AB)

Theoremeqbrtri 3749 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   B𝑅𝐶       A𝑅𝐶

Theoremeqbrtrd 3750 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
(φA = B)    &   (φB𝑅𝐶)       (φA𝑅𝐶)

Theoremeqbrtrri 3751 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   A𝑅𝐶       B𝑅𝐶

Theoremeqbrtrrd 3752 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA = B)    &   (φA𝑅𝐶)       (φB𝑅𝐶)

Theorembreqtri 3753 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A𝑅B    &   B = 𝐶       A𝑅𝐶

Theorembreqtrd 3754 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA𝑅B)    &   (φB = 𝐶)       (φA𝑅𝐶)

Theorembreqtrri 3755 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A𝑅B    &   𝐶 = B       A𝑅𝐶

Theorembreqtrrd 3756 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA𝑅B)    &   (φ𝐶 = B)       (φA𝑅𝐶)

Theorem3brtr3i 3757 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
A𝑅B    &   A = 𝐶    &   B = 𝐷       𝐶𝑅𝐷

Theorem3brtr4i 3758 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
A𝑅B    &   𝐶 = A    &   𝐷 = B       𝐶𝑅𝐷

Theorem3brtr3d 3759 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
(φA𝑅B)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶𝑅𝐷)

Theorem3brtr4d 3760 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
(φA𝑅B)    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐶𝑅𝐷)

Theorem3brtr3g 3761 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(φA𝑅B)    &   A = 𝐶    &   B = 𝐷       (φ𝐶𝑅𝐷)

Theorem3brtr4g 3762 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(φA𝑅B)    &   𝐶 = A    &   𝐷 = B       (φ𝐶𝑅𝐷)

Theoremsyl5eqbr 3763 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
A = B    &   (φB𝑅𝐶)       (φA𝑅𝐶)

Theoremsyl5eqbrr 3764 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
B = A    &   (φB𝑅𝐶)       (φA𝑅𝐶)

Theoremsyl5breq 3765 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
A𝑅B    &   (φB = 𝐶)       (φA𝑅𝐶)

Theoremsyl5breqr 3766 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
A𝑅B    &   (φ𝐶 = B)       (φA𝑅𝐶)

Theoremsyl6eqbr 3767 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
(φA = B)    &   B𝑅𝐶       (φA𝑅𝐶)

Theoremsyl6eqbrr 3768 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
(φB = A)    &   B𝑅𝐶       (φA𝑅𝐶)

Theoremsyl6breq 3769 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
(φA𝑅B)    &   B = 𝐶       (φA𝑅𝐶)

Theoremsyl6breqr 3770 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
(φA𝑅B)    &   𝐶 = B       (φA𝑅𝐶)

Theoremssbrd 3771 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
(φAB)       (φ → (𝐶A𝐷𝐶B𝐷))

Theoremssbri 3772 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
AB       (𝐶A𝐷𝐶B𝐷)

Theoremnfbrd 3773 Deduction version of bound-variable hypothesis builder nfbr 3774. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
(φxA)    &   (φx𝑅)    &   (φxB)       (φ → Ⅎx A𝑅B)

Theoremnfbr 3774 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
xA    &   x𝑅    &   xB       x A𝑅B

Theorembrab1 3775* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
(x𝑅Ax {zz𝑅A})

Theorembrun 3776 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
(A(𝑅𝑆)B ↔ (A𝑅B A𝑆B))

Theorembrin 3777 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
(A(𝑅𝑆)B ↔ (A𝑅B A𝑆B))

Theorembrdif 3778 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
(A(𝑅𝑆)B ↔ (A𝑅B ¬ A𝑆B))

Theoremsbcbrg 3779 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(A 𝐷 → ([A / x]B𝑅𝐶A / xBA / x𝑅A / x𝐶))

Theoremsbcbr12g 3780* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(A 𝐷 → ([A / x]B𝑅𝐶A / xB𝑅A / x𝐶))

Theoremsbcbr1g 3781* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(A 𝐷 → ([A / x]B𝑅𝐶A / xB𝑅𝐶))

Theoremsbcbr2g 3782* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(A 𝐷 → ([A / x]B𝑅𝐶B𝑅A / x𝐶))

2.1.23  Ordered-pair class abstractions (class builders)

Syntaxcopab 3783 Extend class notation to include ordered-pair class abstraction (class builder).
class {⟨x, y⟩ ∣ φ}

Syntaxcmpt 3784 Extend the definition of a class to include maps-to notation for defining a function via a rule.
class (x AB)

Definitiondf-opab 3785* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually x and y are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)
{⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}

Definitiondf-mpt 3786* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from x (in A) to B(x)." The class expression B is the value of the function at x and normally contains the variable x. Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
(x AB) = {⟨x, y⟩ ∣ (x A y = B)}

Theoremopabss 3787* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{⟨x, y⟩ ∣ x𝑅y} ⊆ 𝑅

Theoremopabbid 3788 Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
xφ    &   yφ    &   (φ → (ψχ))       (φ → {⟨x, y⟩ ∣ ψ} = {⟨x, y⟩ ∣ χ})

Theoremopabbidv 3789* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)
(φ → (ψχ))       (φ → {⟨x, y⟩ ∣ ψ} = {⟨x, y⟩ ∣ χ})

Theoremopabbii 3790 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
(φψ)       {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ}

Theoremnfopab 3791* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
zφ       z{⟨x, y⟩ ∣ φ}

Theoremnfopab1 3792 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
x{⟨x, y⟩ ∣ φ}

Theoremnfopab2 3793 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
y{⟨x, y⟩ ∣ φ}

Theoremcbvopab 3794* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
zφ    &   wφ    &   xψ    &   yψ    &   ((x = z y = w) → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨z, w⟩ ∣ ψ}

Theoremcbvopabv 3795* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
((x = z y = w) → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨z, w⟩ ∣ ψ}

Theoremcbvopab1 3796* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
zφ    &   xψ    &   (x = z → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}

Theoremcbvopab2 3797* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
zφ    &   yψ    &   (y = z → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}

Theoremcbvopab1s 3798* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
{⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ [z / x]φ}

Theoremcbvopab1v 3799* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
(x = z → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}

Theoremcbvopab2v 3800* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
(y = z → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}

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 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8569
 Copyright terms: Public domain < Previous  Next >