Home Metamath Proof ExplorerTheorem List (p. 84 of 325) < Previous  Next > Browser slow? Try the Unicode version.

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

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremackm 8301* A remarkable equivalent to the Axiom of Choice that has only 5 quantifiers (when expanded to , primitives in prenex form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by theorem dfackm 8002. Maes found this version of AC in April, 2004 (replacing a longer version, also with 5 quantifiers, that he found in November, 2003). See Kurt Maes, "A 5-quantifier (,=)-expression ZF-equivalent to the Axiom of Choice" (http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf).

The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)

Theoremaxac2 8302* Derive ax-ac2 8299 from ax-ac 8295. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremaxac 8303* Derive ax-ac 8295 from ax-ac2 8299. Note that ax-reg 7516 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)

Theoremaxaci 8304 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremcardeqv 8305 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremnumth3 8306 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremnumth2 8307* Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.)

Theoremnumth 8308* Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)

Theoremac7 8309* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)

Theoremac7g 8310* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)

Theoremac4 8311* Equivalent of Axiom of Choice. We do not insist that be a function. However, theorem ac5 8313, derived from this one, shows that this form of the axiom does imply that at least one such set whose existence we assert is in fact a function. Axiom of Choice of [TakeutiZaring] p. 83.

Takeuti and Zaring call this "weak choice" in contrast to "strong choice" , which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice," Fundamenta Mathematica, 71, 43-62 (1971).

Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 8327. (Contributed by NM, 21-Jul-1996.)

Theoremac4c 8312* Equivalent of Axiom of Choice (class version) (Contributed by NM, 10-Feb-1997.)

Theoremac5 8313* An Axiom of Choice equivalent: there exists a function (called a choice function) with domain that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that be a function is not necessary; see ac4 8311. (Contributed by NM, 29-Aug-1999.)

Theoremac5b 8314* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)

Theoremac6num 8315* A version of ac6 8316 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)

Theoremac6 8316* Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set , where depends on (the natural number) and (to specify a member of ). A stronger version of this theorem, ac6s 8320, allows to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)

Theoremac6c4 8317* Equivalent of Axiom of Choice. is a collection of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)

Theoremac6c5 8318* Equivalent of Axiom of Choice. is a collection of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)

Theoremac9 8319* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.)

Theoremac6s 8320* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 7773, we derive this strong version of ac6 8316 that doesn't require to be a set. (Contributed by NM, 4-Feb-2004.)

Theoremac6n 8321* Equivalent of Axiom of Choice. Contrapositive of ac6s 8320. (Contributed by NM, 10-Jun-2007.)

Theoremac6s2 8322* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8323. (Contributed by NM, 29-Sep-2006.)

Theoremac6s3 8323* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)

Theoremac6sg 8324* ac6s 8320 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)

Theoremac6sf 8325* Version of ac6 8316 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)

Theoremac6s4 8326* Generalization of the Axiom of Choice to proper classes. is a collection of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)

Theoremac6s5 8327* Generalization of the Axiom of Choice to proper classes. is a collection of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)

Theoremac8 8328* An Axiom of Choice equivalent. Given a family of mutually disjoint nonempty sets, there exists a set containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)

Theoremac9s 8329* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes (achieved via the Collection Principle cp 7771). (Contributed by NM, 29-Sep-2006.)

3.2.2  AC equivalents: well-ordering, Zorn's lemma

Theoremnumthcor 8330* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)

Theoremweth 8331* Well-ordering theorem: any set can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario Carneiro, 5-Jan-2013.)

Theoremzorn2lem1 8332* Lemma for zorn2 8342. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem2 8333* Lemma for zorn2 8342. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem3 8334* Lemma for zorn2 8342. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem4 8335* Lemma for zorn2 8342. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem5 8336* Lemma for zorn2 8342. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem6 8337* Lemma for zorn2 8342. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2lem7 8338* Lemma for zorn2 8342. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremzorn2g 8339* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8342 avoids the Axiom of Choice by assuming that is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremzorng 8340* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 8343 avoids the Axiom of Choice by assuming that is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
[]

Theoremzornn0g 8341* Variant of Zorn's lemma zorng 8340 in which , the union of the empty chain, is not required to be an element of . (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
[]

Theoremzorn2 8342* Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set (with an ordering relation ) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 8332 through zorn2lem7 8338; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8338. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremzorn 8343* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 8342 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
[]

Theoremzornn0 8344* Variant of Zorn's lemma zorn 8343 in which , the union of the empty chain, is not required to be an element of . (Contributed by Jeff Madsen, 5-Jan-2011.)
[]

Theoremttukeylem1 8345* Lemma for ttukey 8354. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremttukeylem2 8346* Lemma for ttukey 8354. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremttukeylem3 8347* Lemma for ttukey 8354. (Contributed by Mario Carneiro, 11-May-2015.)
recs

Theoremttukeylem4 8348* Lemma for ttukey 8354. (Contributed by Mario Carneiro, 15-May-2015.)
recs

Theoremttukeylem5 8349* Lemma for ttukey 8354. The function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
recs

Theoremttukeylem6 8350* Lemma for ttukey 8354. (Contributed by Mario Carneiro, 15-May-2015.)
recs

Theoremttukeylem7 8351* Lemma for ttukey 8354. (Contributed by Mario Carneiro, 15-May-2015.)
recs

Theoremttukey2g 8352* The Teichmüller-Tukey Lemma ttukey 8354 with a slightly stronger conclusion: we can set up the maximal element of so that it also contains some given as a subset. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremttukeyg 8353* The Teichmüller-Tukey Lemma ttukey 8354 stated with the "choice" as an antecedent (the hypothesis says that is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.)

Theoremttukey 8354* The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If is a nonempty collection of finite character, then has a maximal element with respect to inclusion. Here "finite character" means that iff every finite subset of is in . (Contributed by Mario Carneiro, 15-May-2015.)

Theoremaxdclem 8355* Lemma for axdc 8357. (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremaxdclem2 8356* Lemma for axdc 8357. Using the full Axiom of Choice, we can construct a choice function on . From this, we can build a sequence starting at any value by repeatedly applying to the set (where is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremaxdc 8357* This theorem derives ax-dc 8282 using ax-ac 8295 and ax-inf 7549. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremfodom 8358 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8310. AC is not needed for finite sets - see fodomfi 7344. See also fodomnum 7894. (Contributed by NM, 23-Jul-2004.)

Theoremfodomg 8359 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)

Theoremfodomb 8360* Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)

Theoremwdomac 8361 When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theorembrdom3 8362* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)

Theorembrdom5 8363* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)

Theorembrdom4 8364* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theorembrdom7disj 8365* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theorembrdom6disj 8366* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)

Theoremfin71ac 8367 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
FinVII

Theoremimadomg 8368 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)

Theoremfnrndomg 8369 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)

Theoremiunfo 8370* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)

Theoremiundom2g 8371* An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. depends on and should be thought of as . (Contributed by Mario Carneiro, 1-Sep-2015.)
AC

Theoremiundomg 8372* An upper bound for the cardinality of an indexed union, with explicit choice principles. depends on and should be thought of as . (Contributed by Mario Carneiro, 1-Sep-2015.)
AC               AC

Theoremiundom 8373* An upper bound for the cardinality of an indexed union. depends on and should be thought of as . (Contributed by NM, 26-Mar-2006.)

Theoremunidom 8374* An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremuniimadom 8375* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)

Theoremuniimadomf 8376* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8375 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)

3.2.3  Cardinal number theorems using Axiom of Choice

Theoremcardval 8377* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 7834 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremcardid 8378 Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremcardidg 8379 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8378. (Contributed by David Moews, 1-May-2017.)

Theoremcardidd 8380 Any set is equinumerous to its cardinal number. Deduction form of cardid 8378. (Contributed by David Moews, 1-May-2017.)

Theoremcardf 8381 The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremcarden 8382 Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank (see karden 7775). (Contributed by NM, 22-Oct-2003.)

Theoremcardeq0 8383 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)

Theoremunsnen 8384 Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)

Theoremcarddom 8385 Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremcardsdom 8386 Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremdomtri 8387 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theorementric 8388 Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.)

Theorementri2 8389 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)

Theorementri3 8390 Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.)

Theoremsdomsdomcard 8391 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)

Theoremcanth3 8392 Cantor's theorem in terms of cardinals. This theorem tells us that no matter how largei a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)

Theoreminfxpidm 8393 The cross product of an infinite set with itself is idempotent. This theorem (which is an AC equivalent) provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This proof follows as a corollary of infxpen 7852. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)

Theoremondomon 8394* The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227. This theorem can be proved (with a longer proof) without the Axiom of Choice; see hartogs 7469. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)

Theoremcardmin 8395* The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.)

Theoremficard 8396 A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminfinf 8397 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)

Theoremunirnfdomd 8398 The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremkonigthlem 8399* Lemma for konigth 8400. (Contributed by Mario Carneiro, 22-Feb-2013.)

Theoremkonigth 8400* Konig's Theorem. If for all , then , where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with regular unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting , this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.)

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 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
 Copyright terms: Public domain < Previous  Next >