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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremral0 3301 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
x φ

Theoremrgenm 3302* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
((x x A x A) → φ)       x A φ

Theoremralf0 3303* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
¬ φ       (x A φA = ∅)

Theoremralm 3304 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
((x x Ax A φ) ↔ x A φ)

Theoremraaanlem 3305* Special case of raaan 3306 where A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
yφ    &   xψ       (x x A → (x A y A (φ ψ) ↔ (x A φ y A ψ)))

Theoremraaan 3306* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
yφ    &   xψ       (x A y A (φ ψ) ↔ (x A φ y A ψ))

Theoremraaanv 3307* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
(x A y A (φ ψ) ↔ (x A φ y A ψ))

Theoremsbss 3308* Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
([y / x]xAyA)

Theoremsbcssg 3309 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
(A 𝑉 → ([A / x]B𝐶A / xBA / x𝐶))

2.1.15  Conditional operator

Syntaxcif 3310 Extend class notation to include the conditional operator. See df-if 3311 for a description. (In older databases this was denoted "ded".)
class if(φ, A, B)

Definitiondf-if 3311* Define the conditional operator. Read if(φ, A, B) as "if φ then A else B." See iftrue 3315 and iffalse 3317 for its values. In mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise."

In the absence of excluded middle, this will tend to be useful where φ is decidable (in the sense of df-dc 734). (Contributed by NM, 15-May-1999.)

if(φ, A, B) = {x ∣ ((x A φ) (x B ¬ φ))}

Theoremdfif6 3312* An alternate definition of the conditional operator df-if 3311 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
if(φ, A, B) = ({x Aφ} ∪ {x B ∣ ¬ φ})

Theoremifeq1 3313 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(A = B → if(φ, A, 𝐶) = if(φ, B, 𝐶))

Theoremifeq2 3314 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(A = B → if(φ, 𝐶, A) = if(φ, 𝐶, B))

Theoremiftrue 3315 Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(φ → if(φ, A, B) = A)

Theoremiftruei 3316 Inference associated with iftrue 3315. (Contributed by BJ, 7-Oct-2018.)
φ       if(φ, A, B) = A

Theoremiffalse 3317 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
φ → if(φ, A, B) = B)

Theoremiffalsei 3318 Inference associated with iffalse 3317. (Contributed by BJ, 7-Oct-2018.)
¬ φ       if(φ, A, B) = B

Theoremifnefalse 3319 When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3317 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
(AB → if(A = B, 𝐶, 𝐷) = 𝐷)

Theoremdfif3 3320* Alternate definition of the conditional operator df-if 3311. Note that φ is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝐶 = {xφ}       if(φ, A, B) = ((A𝐶) ∪ (B ∩ (V ∖ 𝐶)))

Theoremifeq12 3321 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
((A = B 𝐶 = 𝐷) → if(φ, A, 𝐶) = if(φ, B, 𝐷))

Theoremifeq1d 3322 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(φA = B)       (φ → if(ψ, A, 𝐶) = if(ψ, B, 𝐶))

Theoremifeq2d 3323 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(φA = B)       (φ → if(ψ, 𝐶, A) = if(ψ, 𝐶, B))

Theoremifeq12d 3324 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → if(ψ, A, 𝐶) = if(ψ, B, 𝐷))

Theoremifbi 3325 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
((φψ) → if(φ, A, B) = if(ψ, A, B))

Theoremifbid 3326 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
(φ → (ψχ))       (φ → if(ψ, A, B) = if(χ, A, B))

Theoremifbieq1d 3327 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
(φ → (ψχ))    &   (φA = B)       (φ → if(ψ, A, 𝐶) = if(χ, B, 𝐶))

Theoremifbieq2i 3328 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(φψ)    &   A = B       if(φ, 𝐶, A) = if(ψ, 𝐶, B)

Theoremifbieq2d 3329 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(φ → (ψχ))    &   (φA = B)       (φ → if(ψ, 𝐶, A) = if(χ, 𝐶, B))

Theoremifbieq12i 3330 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
(φψ)    &   A = 𝐶    &   B = 𝐷       if(φ, A, B) = if(ψ, 𝐶, 𝐷)

Theoremifbieq12d 3331 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
(φ → (ψχ))    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ → if(ψ, A, B) = if(χ, 𝐶, 𝐷))

Theoremnfifd 3332 Deduction version of nfif 3333. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
(φ → Ⅎxψ)    &   (φxA)    &   (φxB)       (φxif(ψ, A, B))

Theoremnfif 3333 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
xφ    &   xA    &   xB       xif(φ, A, B)

2.1.16  Power classes

Syntaxcpw 3334 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class 𝒫 A

Theorempwjust 3335* Soundness justification theorem for df-pw 3336. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{xxA} = {yyA}

Definitiondf-pw 3336* Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if A is { 3 , 5 , 7 }, then 𝒫 A is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.)
𝒫 A = {xxA}

Theorempweq 3337 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
(A = B → 𝒫 A = 𝒫 B)

Theorempweqi 3338 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
A = B       𝒫 A = 𝒫 B

Theorempweqd 3339 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
(φA = B)       (φ → 𝒫 A = 𝒫 B)

Theoremelpw 3340 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
A V       (A 𝒫 BAB)

Theoremselpw 3341* Setvar variable membership in a power class (common case). See elpw 3340. (Contributed by David A. Wheeler, 8-Dec-2018.)
(x 𝒫 AxA)

Theoremelpwg 3342 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
(A 𝑉 → (A 𝒫 BAB))

Theoremelpwi 3343 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(A 𝒫 BAB)

Theoremelpwid 3344 An element of a power class is a subclass. Deduction form of elpwi 3343. (Contributed by David Moews, 1-May-2017.)
(φA 𝒫 B)       (φAB)

Theoremelelpwi 3345 If A belongs to a part of 𝐶 then A belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
((A B B 𝒫 𝐶) → A 𝐶)

Theoremnfpw 3346 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
xA       x𝒫 A

Theorempwidg 3347 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(A 𝑉A 𝒫 A)

Theorempwid 3348 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
A V       A 𝒫 A

Theorempwss 3349* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 ABx(xAx B))

2.1.17  Unordered and ordered pairs

Syntaxcsn 3350 Extend class notation to include singleton.
class {A}

Syntaxcpr 3351 Extend class notation to include unordered pair.
class {A, B}

Syntaxctp 3352 Extend class notation to include unordered triplet.
class {A, B, 𝐶}

Syntaxcop 3353 Extend class notation to include ordered pair.
class A, B

Syntaxcotp 3354 Extend class notation to include ordered triple.
class A, B, 𝐶

Theoremsnjust 3355* Soundness justification theorem for df-sn 3356. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{xx = A} = {yy = A}

Definitiondf-sn 3356* Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of V, although it is not very meaningful in this case. For an alternate definition see dfsn2 3364. (Contributed by NM, 5-Aug-1993.)
{A} = {xx = A}

Definitiondf-pr 3357 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {A, B} = {B, A} as proven by prcom 3420. For a more traditional definition, but requiring a dummy variable, see dfpr2 3366. (Contributed by NM, 5-Aug-1993.)
{A, B} = ({A} ∪ {B})

Definitiondf-tp 3358 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
{A, B, 𝐶} = ({A, B} ∪ {𝐶})

Definitiondf-op 3359* Definition of an ordered pair, equivalent to Kuratowski's definition {{A}, {A, B}} when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 3545 and opprc2 3546). For Kuratowski's actual definition when the arguments are sets, see dfop 3522.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as A, B⟩ = {{A}, {A, B}}, which has different behavior from our df-op 3359 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3359 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses.

There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition A, B_2 = {{{A}, ∅}, {{B}}}. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is A, B_3 = {A, {A, B}}, but it requires the Axiom of Regularity for its justification and is not commonly used. Finally, an ordered pair of real numbers can be represented by a complex number. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

A, B⟩ = {x ∣ (A V B V x {{A}, {A, B}})}

Definitiondf-ot 3360 Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)
A, B, 𝐶⟩ = ⟨⟨A, B⟩, 𝐶

Theoremsneq 3361 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
(A = B → {A} = {B})

Theoremsneqi 3362 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
A = B       {A} = {B}

Theoremsneqd 3363 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
(φA = B)       (φ → {A} = {B})

Theoremdfsn2 3364 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{A} = {A, A}

Theoremelsn 3365* There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
(x {A} ↔ x = A)

Theoremdfpr2 3366* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{A, B} = {x ∣ (x = A x = B)}

Theoremelprg 3367 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
(A 𝑉 → (A {B, 𝐶} ↔ (A = B A = 𝐶)))

Theoremelpr 3368 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
A V       (A {B, 𝐶} ↔ (A = B A = 𝐶))

Theoremelpr2 3369 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
B V    &   𝐶 V       (A {B, 𝐶} ↔ (A = B A = 𝐶))

Theoremelpri 3370 If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
(A {B, 𝐶} → (A = B A = 𝐶))

Theoremnelpri 3371 If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
AB    &   A𝐶        ¬ A {B, 𝐶}

Theoremelsncg 3372 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A 𝑉 → (A {B} ↔ A = B))

Theoremelsnc 3373 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
A V       (A {B} ↔ A = B)

Theoremelsni 3374 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
(A {B} → A = B)

Theoremsnidg 3375 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
(A 𝑉A {A})

Theoremsnidb 3376 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
(A V ↔ A {A})

Theoremsnid 3377 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
A V       A {A}

Theoremssnid 3378 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
x {x}

Theoremelsnc2g 3379 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 28-Oct-2003.)
(B 𝑉 → (A {B} ↔ A = B))

Theoremelsnc2 3380 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 12-Jun-1994.)
B V       (A {B} ↔ A = B)

Theoremmosn 3381* A singleton has at most one element. This works whether A is a proper class or not, and in that sense can be seen as encompassing both snmg 3460 and snprc 3409. (Contributed by Jim Kingdon, 30-Aug-2018.)
∃*x x {A}

Theoremralsns 3382* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
(A 𝑉 → (x {A}φ[A / x]φ))

Theoremrexsns 3383* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
(x {A}φ[A / x]φ)

TheoremrexsnsOLD 3384* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 3383 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(A 𝑉 → (x {A}φ[A / x]φ))

Theoremralsng 3385* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(x = A → (φψ))       (A 𝑉 → (x {A}φψ))

Theoremrexsng 3386* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
(x = A → (φψ))       (A 𝑉 → (x {A}φψ))

Theoremexsnrex 3387 There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
(x 𝑀 = {x} ↔ x 𝑀 𝑀 = {x})

Theoremralsn 3388* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
A V    &   (x = A → (φψ))       (x {A}φψ)

Theoremrexsn 3389* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
A V    &   (x = A → (φψ))       (x {A}φψ)

Theoremeltpg 3390 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
(A 𝑉 → (A {B, 𝐶, 𝐷} ↔ (A = B A = 𝐶 A = 𝐷)))

Theoremeltpi 3391 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
(A {B, 𝐶, 𝐷} → (A = B A = 𝐶 A = 𝐷))

Theoremeltp 3392 A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)
A V       (A {B, 𝐶, 𝐷} ↔ (A = B A = 𝐶 A = 𝐷))

Theoremdftp2 3393* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)
{A, B, 𝐶} = {x ∣ (x = A x = B x = 𝐶)}

Theoremnfpr 3394 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
xA    &   xB       x{A, B}

Theoremralprg 3395* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(x = A → (φψ))    &   (x = B → (φχ))       ((A 𝑉 B 𝑊) → (x {A, B}φ ↔ (ψ χ)))

Theoremrexprg 3396* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(x = A → (φψ))    &   (x = B → (φχ))       ((A 𝑉 B 𝑊) → (x {A, B}φ ↔ (ψ χ)))

Theoremraltpg 3397* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(x = A → (φψ))    &   (x = B → (φχ))    &   (x = 𝐶 → (φθ))       ((A 𝑉 B 𝑊 𝐶 𝑋) → (x {A, B, 𝐶}φ ↔ (ψ χ θ)))

Theoremrextpg 3398* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
(x = A → (φψ))    &   (x = B → (φχ))    &   (x = 𝐶 → (φθ))       ((A 𝑉 B 𝑊 𝐶 𝑋) → (x {A, B, 𝐶}φ ↔ (ψ χ θ)))

Theoremralpr 3399* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
A V    &   B V    &   (x = A → (φψ))    &   (x = B → (φχ))       (x {A, B}φ ↔ (ψ χ))

Theoremrexpr 3400* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
A V    &   B V    &   (x = A → (φψ))    &   (x = B → (φχ))       (x {A, B}φ ↔ (ψ χ))

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