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Theorem List for Metamath Proof Explorer - 36501-36600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Axiomax-frege31 36501 cannot be denied and (at the same time ) affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnot2 116. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)

Theoremfrege32 36502 Deduce con1 133 from con3 141. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege33 36503 If or takes place, then or takes place. Identical to con1 133. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege34 36504 If as a conseqence of the occurence of the circumstance , when the obstacle is removed, takes place, then from the circumstance that does not take place while occurs the occurence of the obstacle can be inferred. Closed form of con1d 129. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege35 36505 Commuted, closed form of con1d 129. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege36 36506 The case in which is denied, is affirmed, and is affirmed does not occur. If occurs, then (at least) one of the two, or , takes place (no matter what might be). Identical to pm2.24 112. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege37 36507 If is a necessary consequence of the occurrence of or , then is a necessary consequence of alone. Similar to a closed form of orcs 401. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege38 36508 Identical to pm2.21 111. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege39 36509 Syllogism between pm2.18 113 and pm2.24 112. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege40 36510 Anything implies pm2.18 113. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremaxfrege41 36511 Identical to notnot1 126. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)

Axiomax-frege41 36512 The affirmation of denies the denial of . Identical to notnot1 126. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)

Theoremfrege42 36513 Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege43 36514 If there is a choice only between and , then takes place. Identical to pm2.18 113. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege44 36515 Similar to a commuted pm2.62 416. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege45 36516 Deduce pm2.6 175 from con1 133. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege46 36517 If holds when occurs as well as when does not occur, then holds. If or occurs and if the occurences of has as a necessary consequence, then takes place. Identical to pm2.6 175. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege47 36518 Deduce consequence follows from either path implied by a disjunction. If , as well as is sufficient condition for and or takes place, then the proposition holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege48 36519 Closed form of syllogism with internal disjunction. If is a sufficient condition for the occurence of or and if , as well as , is a sufficient condition for , then is a sufficient condition for . See application in frege101 36631. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege49 36520 Closed form of deduction with disjunction. Proposition 49 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege50 36521 Closed form of jaoi 386. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege51 36522 Compare with jaod 387. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

21.25.3.5  _Begriffsschrift_ Chapter II with logical equivalence

Here we leverage df-ifp 984 to partition a wff into two that are disjoint with the selector wff.

Thus if we are given if- then we replace the concept (illegal in our notation ) with if- to reason about the values of the "function." Likewise, we replace the similarly illegal concept with .

Theoremaxfrege52a 36523 Justification for ax-frege52a 36524. (Contributed by RP, 17-Apr-2020.)
if- if-

Axiomax-frege52a 36524 The case when the content of is identical with the content of and in which a proposition controlled by an element for which we substitute the content of is affirmed ( in this specific case the identity logical funtion ) and the same proposition, this time where we subsituted the content of , is denied does not take place. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
if- if-

Theoremfrege52aid 36525 The case when the content of is identical with the content of and in which is affirmed and is denied does not take place. Identical to biimp 198. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege53aid 36526 Specialization of frege53a 36527. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege53a 36527 Lemma for frege55a 36535. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if- if-

Theoremaxfrege54a 36528 Justification for ax-frege54a 36529. Identical to biid 244. (Contributed by RP, 24-Dec-2019.)

Axiomax-frege54a 36529 Reflexive equality of wffs. The content of is identical with the content of . Part of Axiom 54 of [Frege1879] p. 50. Identical to biid 244. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)

Theoremfrege54cor0a 36530 Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-

Theoremfrege54cor1a 36531 Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-

Theoremfrege55aid 36532 Lemma for frege57aid 36539. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.)

Theoremfrege55lem1a 36533 Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-

Theoremfrege55lem2a 36534 Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-

Theoremfrege55a 36535 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-

Theoremfrege55cor1a 36536 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege56aid 36537 Lemma for frege57aid 36539. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege56a 36538 Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if- if- if- if-

Theoremfrege57aid 36539 This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 211. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege57a 36540 Analogue of frege57aid 36539. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if- if-

Theoremaxfrege58a 36541 Identical to anifp 991. Justification for ax-frege58a 36542. (Contributed by RP, 28-Mar-2020.)
if-

Axiomax-frege58a 36542 If is affirmed, cannot be denied. Identical to stdpc4 2202. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
if-

Theoremfrege58acor 36543 Lemma for frege59a 36544. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if- if-

Theoremfrege59a 36544 A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 36480 incorrectly referenced where frege30 36499 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

if- if-

Theoremfrege60a 36545 Swap antecedents of ax-frege58a 36542. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if- if- if-

Theoremfrege61a 36546 Lemma for frege65a 36550. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if-

Theoremfrege62a 36547 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2412 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if- if-

Theoremfrege63a 36548 Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if- if-

Theoremfrege64a 36549 Lemma for frege65a 36550. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if- if- if- if-

Theoremfrege65a 36550 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2412 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if- if-

Theoremfrege66a 36551 Swap antecedents of frege65a 36550. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if- if-

Theoremfrege67a 36552 Lemma for frege68a 36553. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if-

Theoremfrege68a 36553 Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
if-

21.25.3.6  _Begriffsschrift_ Chapter II with equivalence of sets

Theoremaxfrege52c 36554 Justification for ax-frege52c 36555. (Contributed by RP, 24-Dec-2019.)

Axiomax-frege52c 36555 One side of dfsbcq 3257. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)

Theoremfrege52b 36556 The case when the content of is identical with the content of and in which a proposition controlled by an element for which we substitute the content of is affirmed and the same proposition, this time where we subsitute the content of , is denied does not take place. In , can also occur in other than the argument () places. Hence may still be contained in . Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege53b 36557 Lemma for frege102 (via frege92 36622). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremaxfrege54c 36558 Reflexive equality of classes. Identical to eqid 2471. Justification for ax-frege54c 36559. (Contributed by RP, 24-Dec-2019.)

Axiomax-frege54c 36559 Reflexive equality of sets (as classes). Part of Axiom 54 of [Frege1879] p. 50. Identical to eqid 2471. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)

Theoremfrege54b 36560 Reflexive equality of sets. The content of is identical with the content of . Part of Axiom 54 of [Frege1879] p. 50. Slightly specialized version of eqid 2471. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege54cor1b 36561 Reflexive equality. (Contributed by RP, 24-Dec-2019.)

Theoremfrege55lem1b 36562* Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)

Theoremfrege55lem2b 36563 Lemma for frege55b 36564. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege55b 36564 Lemma for frege57b 36566. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2491 incorporates eqcom 2478 which is stronger than this proposition which is identical to equcomi 1869. Is is possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege56b 36565 Lemma for frege57b 36566. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege57b 36566 Analogue of frege57aid 36539. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremaxfrege58b 36567 If is affirmed, cannot be denied. Identical to stdpc4 2202. Justification for ax-frege58b 36568. (Contributed by RP, 28-Mar-2020.)

Axiomax-frege58b 36568 If is affirmed, cannot be denied. Identical to stdpc4 2202. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)

Theoremfrege58bid 36569 If is affirmed, cannot be denied. Identical to sp 1957. See ax-frege58b 36568 and frege58c 36588 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)

Theoremfrege58bcor 36570 Lemma for frege59b 36571. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege59b 36571 A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 36480 incorrectly referenced where frege30 36499 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege60b 36572 Swap antecedents of ax-frege58b 36568. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege61b 36573 Lemma for frege65b 36577. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege62b 36574 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2412 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege63b 36575 Lemma for frege91 36621. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege64b 36576 Lemma for frege65b 36577. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege65b 36577 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2412 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53.

In Frege care is taken to point out that the variables in the first clauses are independent of each other and of the final term so another valid translation could be : . But that is perhaps too pedantic a translation for this exploration. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege66b 36578 Swap antecedents of frege65b 36577. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege67b 36579 Lemma for frege68b 36580. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege68b 36580 Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

21.25.3.7  _Begriffsschrift_ Chapter II with equivalence of classes (where they are sets)

Theoremfrege53c 36581 Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege54cor1c 36582* Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)

Theoremfrege55lem1c 36583* Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)

Theoremfrege55lem2c 36584* Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege55c 36585 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege56c 36586* Lemma for frege57c 36587. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege57c 36587* Swap order of implication in ax-frege52c 36555. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege58c 36588 Principle related to sp 1957. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege59c 36589 A kind of Aristotelian inference. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 36480 incorrectly referenced where frege30 36499 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege60c 36590 Swap antecedents of frege58c 36588. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege61c 36591 Lemma for frege65c 36595. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege62c 36592 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2412 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege63c 36593 Analogue of frege63b 36575. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege64c 36594 Lemma for frege65c 36595. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege65c 36595 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2412 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege66c 36596 Swap antecedents of frege65c 36595. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege67c 36597 Lemma for frege68c 36598. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Theoremfrege68c 36598 Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

21.25.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence

means membership in is hereditary in the sequence dictated by relation . This differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on.

While the above notation is modern, it is cumbersome in the case when is complex and to more closely follow Frege, we abbreviate it with new notation hereditary . This greatly shortens the statements for frege97 36627 and frege109 36639.

dffrege69 36599 through frege75 36605 develop this, but translation to Metamath is pending some decisions.

While Frege does not limit discussion to sets, we may have to depart from Frege by limiting or to sets when we quantify over all hereditary relations or all classes where membership is hereditary in a sequence dictated by .

Theoremdffrege69 36599* If from the proposition that has property it can be inferred generally, whatever may be, that every result of an application of the procedure to has property , then we say " Property is hereditary in the -sequence. Definition 69 of [Frege1879] p. 55. (Contributed by RP, 28-Mar-2020.)
hereditary

Theoremfrege70 36600* Lemma for frege72 36602. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
hereditary

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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 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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 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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