P. 143 of Theorem List - Intuitionistic Logic Explorer

Home	Intuitionistic Logic Explorer Theorem List (p. 143 of 143)	< Previous Wrap >
		Browser slow? Try the Unicode version.
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List

**Theorem List for Intuitionistic Logic Explorer -** 14201-14276 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	nnsf 14201*	Domain and range of . Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)
ℕ_∞ ℕ_∞ℕ_∞

Theorem	peano4nninf 14202*	The successor function on ℕ_∞ is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.)
ℕ_∞ ℕ_∞ℕ_∞

Theorem	peano3nninf 14203*	The successor function on ℕ_∞ is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfalllem1 14204*	Lemma for nninfall 14205. (Contributed by Jim Kingdon, 1-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfall 14205*	Given a decidable predicate on ℕ_∞, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which is a decidable predicate is that it assigns a value of either or (which can be thought of as false and true) to every element of ℕ_∞. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfsellemdc 14206*	Lemma for nninfself 14209. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.)
ℕ_∞ $/\$ DECID

Theorem	nninfsellemcl 14207*	Lemma for nninfself 14209. (Contributed by Jim Kingdon, 8-Aug-2022.)
ℕ_∞ $/\$

Theorem	nninfsellemsuc 14208*	Lemma for nninfself 14209. (Contributed by Jim Kingdon, 6-Aug-2022.)
ℕ_∞ $/\$

Theorem	nninfself 14209*	Domain and range of the selection function for ℕ_∞. (Contributed by Jim Kingdon, 6-Aug-2022.)
ℕ_∞ ℕ_∞ℕ_∞

Theorem	nninfsellemeq 14210*	Lemma for nninfsel 14213. (Contributed by Jim Kingdon, 9-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfsellemqall 14211*	Lemma for nninfsel 14213. (Contributed by Jim Kingdon, 9-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfsellemeqinf 14212*	Lemma for nninfsel 14213. (Contributed by Jim Kingdon, 9-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfsel 14213*	is a selection function for ℕ_∞. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.)
ℕ_∞ ℕ_∞ ℕ_∞

Theorem	nninfomnilem 14214*	Lemma for nninfomni 14215. (Contributed by Jim Kingdon, 10-Aug-2022.)
ℕ_∞ ℕ_∞ Omni

Theorem	nninfomni 14215	ℕ_∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
ℕ_∞ Omni

Theorem	nninffeq 14216*	Equality of two functions on ℕ_∞ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, . (Contributed by Jim Kingdon, 4-Aug-2023.)
ℕ_∞ ℕ_∞

12.3.5 Schroeder-Bernstein Theorem

Theorem	exmidsbthrlem 14217*	Lemma for exmidsbthr 14218. (Contributed by Jim Kingdon, 11-Aug-2022.)
ℕ_∞ $/\$ EXMID

Theorem	exmidsbthr 14218*	The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.)
$/\$ EXMID

Theorem	exmidsbth 14219*	The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6953) is the proof of the Schroeder-Bernstein Theorem from the Metamath Proof Explorer database (in which excluded middle holds), but adapted to use EXMID as an antecedent rather than being unconditionally true, as in the non-intuitionistic proof at https://us.metamath.org/mpeuni/sbth.html 6953. The reverse direction (exmidsbthr 14218) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)
EXMID $/\$

Theorem	sbthomlem 14220	Lemma for sbthom 14221. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.)
Omni ⊔ $\/$

Theorem	sbthom 14221	Schroeder-Bernstein is not possible even for . We know by exmidsbth 14219 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.)
$/\$ $/\$ Omni EXMID

12.3.6 Real and complex numbers

Theorem	qdencn 14222*	The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 11175 (and also would hold for with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
$/\$ $/\$

Theorem	refeq 14223*	Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.)


Theorem	triap 14224	Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
$/\$ $\/$ $\/$ DECID #

Theorem	isomninnlem 14225*	Lemma for isomninn 14226. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.)
frec Omni $\/$

Theorem	isomninn 14226*	Omniscience stated in terms of natural numbers. Similar to isomnimap 7122 but it will sometimes be more convenient to use and rather than and . (Contributed by Jim Kingdon, 30-Aug-2023.)
Omni $\/$

Theorem	cvgcmp2nlemabs 14227*	Lemma for cvgcmp2n 14228. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting as the sum of and a term which gets smaller as gets large. (Contributed by Jim Kingdon, 25-Aug-2023.)
$/\$ $/\$ $/\$

Theorem	cvgcmp2n 14228*	A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.)
$/\$ $/\$ $/\$

Theorem	iooref1o 14229	A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)


Theorem	iooreen 14230	An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)


12.3.7 Analytic omniscience principles Omniscience principles refer to several propositions, most of them weaker than full excluded middle, which do not follow from the axioms of IZF set theory. They are: (0) the Principle of Omniscience (PO), which is another name for excluded middle (see exmidomni 7127), (1) the Limited Principle of Omniscience (LPO) is Omni (see df-omni 7120), (2) the Weak Limited Principle of Omniscience (WLPO) is WOmni (see df-womni 7149), (3) Markov's Principle (MP) is Markov (see df-markov 7137), (4) the Lesser Limited Principle of Omniscience (LLPO) is not yet defined in iset.mm. They also have analytic counterparts each of which follows from the corresponding omniscience principle: (1) Analytic LPO is real number trichotomy, $\/$ $\/$ (see trilpo 14238), (2) Analytic WLPO is decidability of real number equality, DECID (see redcwlpo 14250), (3) Analytic MP is # (see neapmkv 14262), (4) Analytic LLPO is real number dichotomy, $\/$ (most relevant current theorem is maxclpr 11195).

Theorem	trilpolemclim 14231*	Lemma for trilpo 14238. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)


Theorem	trilpolemcl 14232*	Lemma for trilpo 14238. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.)


Theorem	trilpolemisumle 14233*	Lemma for trilpo 14238. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)


Theorem	trilpolemgt1 14234*	Lemma for trilpo 14238. The case. (Contributed by Jim Kingdon, 23-Aug-2023.)


Theorem	trilpolemeq1 14235*	Lemma for trilpo 14238. The case. This is proved by noting that if any is zero, then the infinite sum is less than one based on the term which is zero. We are using the fact that the sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.)


Theorem	trilpolemlt1 14236*	Lemma for trilpo 14238. The case. We can use the distance between and one (that is, ) to find a position in the sequence where terms after that point will not add up to as much as . By finomni 7125 we know the terms up to either contain a zero or are all one. But if they are all one that contradicts the way we constructed , so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.)


Theorem	trilpolemres 14237*	Lemma for trilpo 14238. The result. (Contributed by Jim Kingdon, 23-Aug-2023.)
$\/$ $\/$ $\/$

Theorem	trilpo 14238*	Real number trichotomy implies the Limited Principle of Omniscience (LPO). We expect that we'd need some form of countable choice to prove the converse. Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 14236 (which means the sequence contains a zero), trilpolemeq1 14235 (which means the sequence is all ones), and trilpolemgt1 14234 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 14224) or that the real numbers are a discrete field (see trirec0 14239). LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10208 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)
$\/$ $\/$ Omni

Theorem	trirec0 14239*	Every real number having a reciprocal or equaling zero is equivalent to real number trichotomy. This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 14238). (Contributed by Jim Kingdon, 10-Jun-2024.)
$\/$ $\/$ $\/$

Theorem	trirec0xor 14240*	Version of trirec0 14239 with exclusive-or. The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.)
$\/$ $\/$ $\/_$

Theorem	apdifflemf 14241	Lemma for apdiff 14243. Being apart from the point halfway between and suffices for to be a different distance from and from . (Contributed by Jim Kingdon, 18-May-2024.)
# #

Theorem	apdifflemr 14242	Lemma for apdiff 14243. (Contributed by Jim Kingdon, 19-May-2024.)
# $/\$ # #

Theorem	apdiff 14243*	The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.)
# #

Theorem	iswomninnlem 14244*	Lemma for iswomnimap 7151. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.)
frec WOmni DECID

Theorem	iswomninn 14245*	Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7151 but it will sometimes be more convenient to use and rather than and . (Contributed by Jim Kingdon, 20-Jun-2024.)
WOmni DECID

Theorem	iswomni0 14246*	Weak omniscience stated in terms of equality with . Like iswomninn 14245 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)
WOmni DECID

Theorem	ismkvnnlem 14247*	Lemma for ismkvnn 14248. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
frec Markov

Theorem	ismkvnn 14248*	The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.)
Markov

Theorem	redcwlpolemeq1 14249*	Lemma for redcwlpo 14250. A biconditionalized version of trilpolemeq1 14235. (Contributed by Jim Kingdon, 21-Jun-2024.)


Theorem	redcwlpo 14250*	Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse. Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 14249). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones. Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO". WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10212 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)
DECID WOmni

Theorem	tridceq 14251*	Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14238 and redcwlpo 14250). Thus, this is an analytic analogue to lpowlpo 7153. (Contributed by Jim Kingdon, 24-Jul-2024.)
$\/$ $\/$ DECID

Theorem	redc0 14252*	Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
DECID DECID

Theorem	reap0 14253*	Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
$\/$ $\/$ DECID #

Theorem	dceqnconst 14254*	Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 14250 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)
DECID $/\$ $/\$

Theorem	dcapnconst 14255*	Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See trilpo 14238 for more discussion of decidability of real number apartness. This is a weaker form of dceqnconst 14254 and in fact this theorem can be proved using dceqnconst 14254 as shown at dcapnconstALT 14256. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)
DECID # $/\$ $/\$

Theorem	dcapnconstALT 14256*	Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 14255 by means of dceqnconst 14254. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
DECID # $/\$ $/\$

Theorem	nconstwlpolem0 14257*	Lemma for nconstwlpo 14260. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.)


Theorem	nconstwlpolemgt0 14258*	Lemma for nconstwlpo 14260. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)


Theorem	nconstwlpolem 14259*	Lemma for nconstwlpo 14260. (Contributed by Jim Kingdon, 23-Jul-2024.)
$/\$ $\/$

Theorem	nconstwlpo 14260*	Existence of a certain non-constant function from reals to integers implies WOmni (the Weak Limited Principle of Omniscience or WLPO). Based on Exercise 11.6(ii) of [HoTT], p. (varies). (Contributed by BJ and Jim Kingdon, 22-Jul-2024.)
$/\$ WOmni

Theorem	neapmkvlem 14261*	Lemma for neapmkv 14262. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
$/\$ #

Theorem	neapmkv 14262*	If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.)
# Markov

12.3.8 Supremum and infimum

Theorem	supfz 14263	The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)


Theorem	inffz 14264	The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
inf

12.3.9 Circle constant

Theorem	taupi 14265	Relationship between and . This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)


12.4 Mathbox for Mykola Mostovenko

Theorem	ax1hfs 14266	Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.)
$/\$

12.5 Mathbox for David A. Wheeler

12.5.1 Testable propositions

Theorem	dftest 14267	A proposition is testable iff its negative or double-negative is true. See Chapter 2 [Moschovakis] p. 2. We do not formally define testability with a new token, but instead use DECID before the formula in question. For example, DECID corresponds to " is testable". (Contributed by David A. Wheeler, 13-Aug-2018.) For statements about testable propositions, search for the keyword "testable" in the comments of statements, for instance using the Metamath command "MM> SEARCH * "testable" / COMMENTS". (New usage is discouraged.)
DECID $\/$

12.5.2 Allsome quantifier These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like do not imply that is ever true, leading to vacuous truths. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ! , and when restricted (applied to a class) we allow ! . The first symbol after the setvar variable must always be if it is the form applied to a class, and since cannot begin a wff, it is unambiguous. The looks like it would be a problem because or might include implications, but any implication arrow within any wff must be surrounded by parentheses, so only the implication arrow of ! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome.

Syntax	walsi 14268	Extend wff definition to include "all some" applied to a top-level implication, which means is true whenever is true, and there is at least least one where is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
!

Syntax	walsc 14269	Extend wff definition to include "all some" applied to a class, which means is true for all in , and there is at least one in . (Contributed by David A. Wheeler, 20-Oct-2018.)
!

Definition	df-alsi 14270	Define "all some" applied to a top-level implication, which means is true whenever is true and there is at least one where is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
! $/\$

Definition	df-alsc 14271	Define "all some" applied to a class, which means is true for all in and there is at least one in . (Contributed by David A. Wheeler, 20-Oct-2018.)
! $/\$

Theorem	alsconv 14272	There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
! !

Theorem	alsi1d 14273	Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
!

Theorem	alsi2d 14274	Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
!

Theorem	alsc1d 14275	Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
!

Theorem	alsc2d 14276	Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
!

< Previous Wrap >

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 14201-14276

< Previous Wrap >