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Theorem List for Metamath Proof Explorer - 38001-38100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelexpmulnn 38001 With exponents limited to the counting numbers, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  ( J  x.  K ) )  /\  ( J  e.  NN  /\  K  e.  NN ) )  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremrelexpmulg 38002 With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  ( J  x.  K ) 
 /\  ( I  =  0  ->  J  <_  K ) )  /\  ( J  e.  NN0  /\  K  e.  NN0 ) )  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremtrclrelexplem 38003* The union of relational powers to positive multiples of  N is a subset to the transitive closure raised to the power of  N. (Contributed by RP, 15-Jun-2020.)
 |-  ( N  e.  NN  ->  U_ k  e.  NN  (
 ( D ^r 
 k ) ^r  N )  C_  ( U_ j  e.  NN  ( D ^r  j ) ^r  N ) )
 
Theoremiunrelexpmin2 38004* The indexed union of relation exponentiation over the natural numbers (including zero) is the minimum reflexive-transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   =>    |-  ( ( R  e.  V  /\  N  =  NN0 )  ->  A. s ( ( (  _I  |`  ( dom 
 R  u.  ran  R ) )  C_  s  /\  R  C_  s  /\  (
 s  o.  s ) 
 C_  s )  ->  ( C `  R ) 
 C_  s ) )
 
Theoremrelexp01min 38005 With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)
 |-  (
 ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
 )  /\  ( J  e.  { 0 ,  1 }  /\  K  e.  { 0 ,  1 } ) )  ->  (
 ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
 
Theoremrelexp1idm 38006 Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
 |-  ( R  e.  V  ->  ( ( R ^r 
 1 ) ^r 
 1 )  =  ( R ^r  1 ) )
 
Theoremrelexp0idm 38007 Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
 |-  ( R  e.  V  ->  ( ( R ^r 
 0 ) ^r 
 0 )  =  ( R ^r  0 ) )
 
Theoremrelexp0a 38008 Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
 |-  (
 ( A  e.  V  /\  N  e.  NN0 )  ->  ( ( A ^r  N ) ^r 
 0 )  C_  ( A ^r  0 ) )
 
Theoremrelexpxpmin 38009 The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  /\  ( I  =  if ( J  <  K ,  J ,  K )  /\  J  e.  NN0  /\  K  e.  NN0 ) )  ->  (
 ( ( A  X.  B ) ^r  J ) ^r  K )  =  (
 ( A  X.  B ) ^r  I ) )
 
Theoremrelexpaddss 38010 The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where  R is a relation as shown by relexpaddd 13794 or when the sum of the powers isn't 1 as shown by relexpaddg 13793. (Contributed by RP, 3-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  M  e.  NN0  /\  R  e.  V )  ->  (
 ( R ^r  N )  o.  ( R ^r  M ) )  C_  ( R ^r  ( N  +  M ) ) )
 
Theoremiunrelexpuztr 38011* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 13800. (Contributed by RP, 4-Jun-2020.)
 |-  C  =  ( r  e.  _V  |->  U_ n  e.  N  ( r ^r  n ) )   =>    |-  ( ( R  e.  V  /\  N  =  (
 ZZ>= `  M )  /\  M  e.  NN0 )  ->  ( ( C `  R )  o.  ( C `  R ) ) 
 C_  ( C `  R ) )
 
20.27.2.4  Transitive closure of a relation
 
Theoremdftrcl3 38012* Transitive closure of a relation, expressed as indexed union of powers of relations. (Contributed by RP, 5-Jun-2020.)
 |-  t+  =  ( r  e.  _V  |->  U_ n  e.  NN  ( r ^r  n ) )
 
Theorembrfvtrcld 38013* If two elements are connected by the transitive closure of a relation, then they are connected via 
n instances the relation, for some counting number  n. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( t+ `
  R ) B  <->  E. n  e.  NN  A ( R ^r  n ) B ) )
 
Theoremfvtrcllb1d 38014 A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( t+ `  R ) )
 
Theoremtrclfvcom 38015 The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.)
 |-  ( R  e.  V  ->  ( ( t+ `  R )  o.  R )  =  ( R  o.  ( t+ `  R ) ) )
 
Theoremcnvtrclfv 38016 The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020.)
 |-  ( R  e.  V  ->  `' ( t+ `  R )  =  (
 t+ `  `' R ) )
 
Theoremcotrcltrcl 38017 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
 |-  (
 t+  o.  t+ )  =  t+
 
Theoremtrclimalb2 38018 Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
 |-  (
 ( R  e.  V  /\  ( R " ( A  u.  B ) ) 
 C_  B )  ->  ( ( t+ `
  R ) " A )  C_  B )
 
Theorembrtrclfv2 38019* Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
 |-  (
 ( X  e.  U  /\  Y  e.  V  /\  R  e.  W )  ->  ( X ( t+ `  R ) Y  <->  Y  e.  |^| { f  |  ( R " ( { X }  u.  f
 ) )  C_  f } ) )
 
Theoremtrclfvdecomr 38020 The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
 |-  ( R  e.  V  ->  ( t+ `  R )  =  ( R  u.  ( ( t+ `
  R )  o.  R ) ) )
 
Theoremtrclfvdecoml 38021 The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
 |-  ( R  e.  V  ->  ( t+ `  R )  =  ( R  u.  ( R  o.  (
 t+ `  R ) ) ) )
 
TheoremdmtrclfvRP 38022 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
 |-  ( R  e.  V  ->  dom  ( t+ `  R )  =  dom  R )
 
TheoremrntrclfvRP 38023 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 19-Jul-2020.) (Proof modification is discouraged.)
 |-  ( R  e.  V  ->  ran  ( t+ `  R )  =  ran  R )
 
Theoremrntrclfv 38024 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
 |-  ( R  e.  V  ->  ran  ( t+ `  R )  =  ran  R )
 
Theoremdfrtrcl3 38025* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 13802. (Contributed by RP, 5-Jun-2020.)
 |-  t*  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r  n ) )
 
Theorembrfvrtrcld 38026* If two elements are connected by the reflexive-transitive closure of a relation, then they are connected via  n instances the relation, for some natural number  n. Similar of dfrtrclrec2 13797. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( t* `
  R ) B  <->  E. n  e.  NN0  A ( R ^r  n ) B ) )
 
Theoremfvrtrcllb0d 38027 A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R ) ) 
 C_  ( t* `
  R ) )
 
Theoremfvrtrcllb0da 38028 A restriction of the identity relation is a subset of the reflexive-transitive closure of a relation. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  U. U. R )  C_  ( t* `
  R ) )
 
Theoremfvrtrcllb1d 38029 A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( t* `  R ) )
 
Theoremdfrtrcl4 38030 Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.)
 |-  t*  =  ( r  e.  _V  |->  ( ( r ^r  0 )  u.  ( t+ `
  r ) ) )
 
Theoremcorcltrcl 38031 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
 |-  (
 r*  o.  t+ )  =  t*
 
Theoremcortrcltrcl 38032 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  t+ )  =  t*
 
Theoremcorclrtrcl 38033 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 r*  o.  t* )  =  t*
 
Theoremcotrclrcl 38034 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
 |-  (
 t+  o.  r* )  =  t*
 
Theoremcortrclrcl 38035 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  r* )  =  t*
 
Theoremcotrclrtrcl 38036 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t+  o.  t* )  =  t*
 
Theoremcortrclrtrcl 38037 The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  t* )  =  t*
 
20.27.2.5  Adapted from Frege

Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879].

 
Theoremfrege77d 38038 If the images of both  { A } and  U are subsets of  U and  B follows  A in the transitive closure of  R, then  B is an element of  U. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 38234. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  A ( t+ `  R ) B )   &    |-  ( ph  ->  ( R " U ) 
 C_  U )   &    |-  ( ph  ->  ( R " { A } )  C_  U )   =>    |-  ( ph  ->  B  e.  U )
 
Theoremfrege81d 38039 If the image of  U is a subset  U,  A is an element of  U and  B follows  A in the transitive closure of  R, then  B is an element of  U. Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 38238. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  A ( t+ `  R ) B )   &    |-  ( ph  ->  ( R " U ) 
 C_  U )   =>    |-  ( ph  ->  B  e.  U )
 
Theoremfrege83d 38040 If the image of the union of  U and  V is a subset of the union of  U and  V,  A is an element of  U and  B follows  A in the transitive closure of 
R, then  B is an element of the union of  U and  V. Similar to Proposition 83 of [Frege1879] p. 65. Compare with frege83 38240. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  A ( t+ `  R ) B )   &    |-  ( ph  ->  ( R " ( U  u.  V ) ) 
 C_  ( U  u.  V ) )   =>    |-  ( ph  ->  B  e.  ( U  u.  V ) )
 
Theoremfrege96d 38041 If  C follows  A in the transitive closure of  R and  B follows  C in  R, then  B follows  A in the transitive closure of  R. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 38253. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  A ( t+ `  R ) C )   &    |-  ( ph  ->  C R B )   =>    |-  ( ph  ->  A ( t+ `  R ) B )
 
Theoremfrege87d 38042 If the images of both  { A } and  U are subsets of  U and  C follows  A in the transitive closure of  R and  B follows  C in  R, then  B is an element of  U. Similar to Proposition 87 of [Frege1879] p. 66. Compare with frege87 38244. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  A ( t+ `  R ) C )   &    |-  ( ph  ->  C R B )   &    |-  ( ph  ->  ( R " { A } )  C_  U )   &    |-  ( ph  ->  ( R " U )  C_  U )   =>    |-  ( ph  ->  B  e.  U )
 
Theoremfrege91d 38043 If  B follows  A in  R then  B follows  A in the transitive closure of  R. Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 38248. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  A ( t+ `  R ) B )
 
Theoremfrege97d 38044 If  A contains all elements after those in  U in the transitive closure of  R, then the image under  R of  A is a subclass of  A. Similar to Proposition 97 of [Frege1879] p. 71. Compare with frege97 38254. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  =  ( ( t+ `
  R ) " U ) )   =>    |-  ( ph  ->  ( R " A ) 
 C_  A )
 
Theoremfrege98d 38045 If  C follows  A and  B follows  C in the transitive closure of  R, then  B follows  A in the transitive closure of  R. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 38255. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  A ( t+ `  R ) C )   &    |-  ( ph  ->  C ( t+ `  R ) B )   =>    |-  ( ph  ->  A (
 t+ `  R ) B )
 
Theoremfrege102d 38046 If either  A and  C are the same or  C follows  A in the transitive closure of  R and  B is the successor to  C, then  B follows  A in the transitive closure of  R. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 38259. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  ( A ( t+ `
  R ) C  \/  A  =  C ) )   &    |-  ( ph  ->  C R B )   =>    |-  ( ph  ->  A ( t+ `  R ) B )
 
Theoremfrege106d 38047 If  B follows  A in  R, then either  A and 
B are the same or  B follows  A in  R. Similar to Proposition 106 of [Frege1879] p. 73. Compare with frege106 38263. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( A R B  \/  A  =  B ) )
 
Theoremfrege108d 38048 If either  A and  C are the same or  C follows  A in the transitive closure of  R and  B is the successor to  C, then either  A and  B are the same or  B follows  A in the transitive closure of  R. Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 38265. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  ( A ( t+ `
  R ) C  \/  A  =  C ) )   &    |-  ( ph  ->  C R B )   =>    |-  ( ph  ->  ( A ( t+ `
  R ) B  \/  A  =  B ) )
 
Theoremfrege109d 38049 If  A contains all elements of  U and all elements after those in  U in the transitive closure of  R, then the image under  R of  A is a subclass of  A. Similar to Proposition 109 of [Frege1879] p. 74. Compare with frege109 38266. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  =  ( U  u.  (
 ( t+ `  R ) " U ) ) )   =>    |-  ( ph  ->  ( R " A ) 
 C_  A )
 
Theoremfrege114d 38050 If either  R relates  A and  B or  A and  B are the same, then either  A and  B are the same,  R relates  A and  B,  R relates  B and  A. Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 38271. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  ( A R B  \/  A  =  B ) )   =>    |-  ( ph  ->  ( A R B  \/  A  =  B  \/  B R A ) )
 
Theoremfrege111d 38051 If either  A and  C are the same or  C follows  A in the transitive closure of  R and  B is the successor to  C, then either  A and  B are the same or  A follows  B or  B and  A in the transitive closure of  R. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 38268. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  ( A ( t+ `
  R ) C  \/  A  =  C ) )   &    |-  ( ph  ->  C R B )   =>    |-  ( ph  ->  ( A ( t+ `
  R ) B  \/  A  =  B  \/  B ( t+ `
  R ) A ) )
 
Theoremfrege122d 38052 If  F is a function,  A is the successor of  X, and  B is the successor of  X, then  A and  B are the same (or  B follows  A in the transitive closure of  F). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 38279. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  A  =  ( F `  X ) )   &    |-  ( ph  ->  B  =  ( F `  X ) )   =>    |-  ( ph  ->  ( A ( t+ `
  F ) B  \/  A  =  B ) )
 
Theoremfrege124d 38053 If  F is a function,  A is the successor of  X, and  B follows  X in the transitive closure of  F, then  A and  B are the same or  B follows  A in the transitive closure of  F. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 38281. (Contributed by RP, 16-Jul-2020.)
 |-  ( ph  ->  F  e.  _V )   &    |-  ( ph  ->  X  e.  dom  F )   &    |-  ( ph  ->  A  =  ( F `  X ) )   &    |-  ( ph  ->  X ( t+ `  F ) B )   &    |-  ( ph  ->  Fun  F )   =>    |-  ( ph  ->  ( A ( t+ `  F ) B  \/  A  =  B )
 )
 
Theoremfrege126d 38054 If  F is a function,  A is the successor of  X, and  B follows  X in the transitive closure of  F, then (for distinct  A and  B) either  A follows  B or  B follows  A in the transitive closure of  F. Similar to Proposition 126 of [Frege1879] p. 81. Compare with frege126 38283. (Contributed by RP, 16-Jul-2020.)
 |-  ( ph  ->  F  e.  _V )   &    |-  ( ph  ->  X  e.  dom  F )   &    |-  ( ph  ->  A  =  ( F `  X ) )   &    |-  ( ph  ->  X ( t+ `  F ) B )   &    |-  ( ph  ->  Fun  F )   =>    |-  ( ph  ->  ( A ( t+ `  F ) B  \/  A  =  B  \/  B ( t+ `
  F ) A ) )
 
Theoremfrege129d 38055 If  F is a function and (for distinct  A and  B) either  A follows  B or  B follows  A in the transitive closure of  F, the successor of  A is either  B or it follows  B or it comes before  B in the transitive closure of  F. Similar to Proposition 129 of [Frege1879] p. 83. Comparw with frege129 38286. (Contributed by RP, 16-Jul-2020.)
 |-  ( ph  ->  F  e.  _V )   &    |-  ( ph  ->  A  e.  dom  F )   &    |-  ( ph  ->  C  =  ( F `  A ) )   &    |-  ( ph  ->  ( A ( t+ `
  F ) B  \/  A  =  B  \/  B ( t+ `
  F ) A ) )   &    |-  ( ph  ->  Fun 
 F )   =>    |-  ( ph  ->  ( B ( t+ `
  F ) C  \/  B  =  C  \/  C ( t+ `
  F ) B ) )
 
Theoremfrege131d 38056 If  F is a function and  A contains all elements of  U and all elements before or after those elements of  U in the transitive closure of  F, then the image under  F of  A is a subclass of  A. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 38288. (Contributed by RP, 17-Jul-2020.)
 |-  ( ph  ->  F  e.  _V )   &    |-  ( ph  ->  A  =  ( U  u.  (
 ( `' ( t+ `  F )
 " U )  u.  ( ( t+ `
  F ) " U ) ) ) )   &    |-  ( ph  ->  Fun 
 F )   =>    |-  ( ph  ->  ( F " A )  C_  A )
 
Theoremfrege133d 38057 If  F is a function and  A and  B both follow  X in the transitive closure of  F, then (for distinct  A and  B) either  A follows  B or  B follows  A in the transitive closure of  F (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 38290. (Contributed by RP, 18-Jul-2020.)
 |-  ( ph  ->  F  e.  _V )   &    |-  ( ph  ->  X ( t+ `  F ) A )   &    |-  ( ph  ->  X (
 t+ `  F ) B )   &    |-  ( ph  ->  Fun 
 F )   =>    |-  ( ph  ->  ( A ( t+ `
  F ) B  \/  A  =  B  \/  B ( t+ `
  F ) A ) )
 
20.27.3  Propositions from _Begriffsschrift_

In 1879, Frege introduced notation for documenting formal reasoning about propositions (and classes) which covered elements of propositional logic, predicate calculus and reasoning about relations. However, due to the pitfalls of naive set theory, adapting this work for inclusion in set.mm required dividing statements about propositions from those about classes and identifying when a restriction to sets is required. For an overview comparing the details of Frege's two-dimensional notation and that used in set.mm, see mmfrege.html. See ru 3434 for discussion of an example of a class that is not a set.

Numbered propositions from [Frege1879]. ax-frege1 38084, ax-frege2 38085, ax-frege8 38103, ax-frege28 38124, ax-frege31 38128, ax-frege41 38139, frege52 (see ax-frege52a 38151, frege52b 38183, and ax-frege52c 38182 for translations), frege54 (see ax-frege54a 38156, frege54b 38187 and ax-frege54c 38186 for translations) and frege58 (see ax-frege58a 38169, ax-frege58b 38195 and frege58c 38215 for translations) are considered "core" or axioms. However, at least ax-frege8 38103 can be derived from ax-frege1 38084 and ax-frege2 38085, see axfrege8 38101.

Frege introduced implication, negation and the universal qualifier as primitives and did not in the numbered propositions use other logical connectives other than equivalence introduced in ax-frege52a 38151, frege52b 38183, and ax-frege52c 38182. In dffrege69 38226, Frege introduced  R hereditary  A to say that relation  R, when restricted to operate on elements of class  A, will only have elements of class  A in its domain; see df-he 38067 for a definition in terms of image and subset. In dffrege76 38233, Frege introduced notation for the concept of two sets related by the transitive closure of a relation, for which we write  X (
t+ `  R
) Y, which requires  R to also be a set. In dffrege99 38256, Frege introduced notation for the concept of two sets either identical or related by the transitive closure of a relation, for which we write  X ( ( t+ `  R
)  u.  _I  ) Y, which is a superclass of sets related by the reflexive-transitive relation  X
( t* `  R ) Y. Finally, in dffrege115 38272, Frege introduced notation for the concept of a relation having the property elements in its domain pair up with only one element each in its range, for which we write  Fun  `' `' R (to ignore any non-relational content of the class  R). Frege did this without the expressing concept of a relation (or its transitive closure) as a class, and needed to invent conventions for discussing indeterminate propositions with two slots free and how to recognize which of the slots was domain and which was range. See mmfrege.html for details.

English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger set.mm database has also been made. See frege77d 38038 for an example.

 
20.27.3.1  _Begriffsschrift_ Chapter I

Section 2 introduces the turnstile  |- which turns an idea which may be true  ph into an assertion that it does hold true  |- 
ph. Section 5 introduces implication, 
( ph  ->  ps ). Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or  ( -.  ph  ->  ps ), and  -.  ( ph  ->  -.  ps ), and two for exclusive-or corresponding to df-or 385, df-an 386, dfxor4 38058, dfxor5 38059.

Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biimplication  ( ph  <->  ps ) or class equality  A  =  B in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f( ph) is interpreted to mean if- ( ph ,  ps ,  ch ) where the content of the "function" is specified by the latter two argments or logical equivalent, while g( A) is read as  A  e.  G and h( A ,  B) as  A H B. This necessarily introduces a need for set theory as both  A  e.  G and  A H B cannot hold unless  A is a set. (Also  B.)

Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f( ph) as if- ( ph ,  ps ,  ch ) would result in the translation of  A. ph f ( ph) as  ( ps 
/\  ch ). For collections, we must generalize over set variables or run into the same problems; this leads to  A. A g( A) being translated as  A. a a  e.  G and so forth.

Under this interpreation the text of section 11 gives us sp 2053 (or simpl 473 and simpr 477 and anifp 1020 in the propositional case) and statments similar to cbvalivw 1934, ax-gen 1722, alrimiv 1855, and alrimdv 1857. These last four introduce a generality and have no useful definition in terms of propositional variables.

Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A,  A. x x  e.  A,  -.  E. x -.  x  e.  A alex 1753, 
A  =  _V eqv 3205; Some are not B,  -.  A. x x  e.  B,  E. x -.  x  e.  B exnal 1754, 
B  C.  _V pssv 4016,  B  =/=  _V nev 38062; There are no C,  A. x -.  x  e.  C,  -.  E. x x  e.  C alnex 1706, 
C  =  (/) eq0 3929; There exist D,  -. 
A. x -.  x  e.  D,  E. x x  e.  D df-ex 1705,  (/)  C.  D 0pss 4013,  D  =/=  (/) n0 3931.

Notation for relations between expressions also can be written in various ways. All E are P,  A. x ( x  e.  E  ->  x  e.  P ),  -.  E. x
( x  e.  E  /\  -.  x  e.  P
) dfss6 3593, 
E  =  ( E  i^i  P ) df-ss 3588,  E  C_  P dfss2 3591; No F are P,  A. x ( x  e.  F  ->  -.  x  e.  P ),  -.  E. x
( x  e.  F  /\  x  e.  P
) alinexa 1770,  ( F  i^i  P
)  =  (/) disj1 4019; Some G are not P,  -.  A. x ( x  e.  G  ->  x  e.  P ),  E. x ( x  e.  G  /\  -.  x  e.  P
) exanali 1786,  ( G  i^i  P
)  C.  G nssinpss 3856,  -.  G  C_  P nss 3663; Some H are P,  -.  A. x
( x  e.  H  ->  -.  x  e.  P
),  E. x ( x  e.  H  /\  x  e.  P ) bj-exnalimn 32610,  (/)  C.  ( H  i^i  P
) 0pssin 38064, 
( H  i^i  P
)  =/=  (/) ndisj 38063.

 
Theoremdfxor4 38058 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  \/_  ps )  <->  -.  ( ( -.  ph  ->  ps )  ->  -.  ( ph  ->  -.  ps )
 ) )
 
Theoremdfxor5 38059 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
 |-  (
 ( ph  \/_  ps )  <->  -.  ( ( ph  ->  -. 
 ps )  ->  -.  ( -.  ph  ->  ps )
 ) )
 
Theoremdf3or2 38060 Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)
 |-  (
 ( ph  \/  ps  \/  ch )  <->  ( -.  ph  ->  ( -.  ps  ->  ch ) ) )
 
Theoremdf3an2 38061 Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)
 |-  (
 ( ph  /\  ps  /\  ch )  <->  -.  ( ph  ->  ( ps  ->  -.  ch )
 ) )
 
Theoremnev 38062* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  =/=  _V  <->  -.  A. x  x  e.  A )
 
Theoremndisj 38063* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
 |-  (
 ( A  i^i  B )  =/=  (/)  <->  E. x ( x  e.  A  /\  x  e.  B ) )
 
Theorem0pssin 38064* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
 |-  ( (/)  C.  ( A  i^i  B ) 
 <-> 
 E. x ( x  e.  A  /\  x  e.  B ) )
 
20.27.3.2  _Begriffsschrift_ Notation hints

The statement  R hereditary  A means relation  R is hereditary (in the sense of Frege) in the class  A or  ( R " A )  C_  A. The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked.

As Frege was not using the language of classes or sets, this naturally 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.

 
Theoremrp-imass 38065 If the  R-image of a class  A is a subclass of  B, then the restriction of  R to  A is a subset of the Cartesian product of  A and  B. (Contributed by Richard Penner, 24-Dec-2019.)
 |-  (
 ( R " A )  C_  B  <->  ( R  |`  A ) 
 C_  ( A  X.  B ) )
 
Syntaxwhe 38066 The property of relation  R being hereditary in class  A.
 wff  R hereditary  A
 
Definitiondf-he 38067 The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
 |-  ( R hereditary  A  <->  ( R " A )  C_  A )
 
Theoremdfhe2 38068 The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
 |-  ( R hereditary  A  <->  ( R  |`  A ) 
 C_  ( A  X.  A ) )
 
Theoremdfhe3 38069* The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
 |-  ( R hereditary  A  <->  A. x ( x  e.  A  ->  A. y
 ( x R y 
 ->  y  e.  A ) ) )
 
Theoremheeq12 38070 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  (
 ( R  =  S  /\  A  =  B ) 
 ->  ( R hereditary  A  <->  S hereditary  B ) )
 
Theoremheeq1 38071 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( R  =  S  ->  ( R hereditary  A  <->  S hereditary  A ) )
 
Theoremheeq2 38072 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( A  =  B  ->  ( R hereditary  A  <->  R hereditary  B ) )
 
Theoremsbcheg 38073 Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ]. B hereditary  C  <->  [_ A  /  x ]_ B hereditary  [_ A  /  x ]_ C ) )
 
Theoremhess 38074 Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( S  C_  R  ->  ( R hereditary  A  ->  S hereditary  A ) )
 
Theoremxphe 38075 Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
 |-  ( A  X.  B ) hereditary  B
 
Theorem0he 38076 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
 |-  (/) hereditary  A
 
Theorem0heALT 38077 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (/) hereditary  A
 
Theoremhe0 38078 Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
 |-  A hereditary  (/)
 
Theoremunhe1 38079 The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
 |-  (
 ( R hereditary  A  /\  S hereditary  A )  ->  ( R  u.  S ) hereditary  A )
 
Theoremsnhesn 38080 Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
 |-  { <. A ,  A >. } hereditary  { B }
 
Theoremidhe 38081 The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
 |-  _I hereditary  A
 
Theorempsshepw 38082 The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
 |-  `' [ C.] hereditary  ~P A
 
Theoremsshepw 38083 The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
 |-  ( `' [ C.]  u.  _I  ) hereditary  ~P A
 
20.27.3.3  _Begriffsschrift_ Chapter II Implication
 
Axiomax-frege1 38084 The case in which  ph is denied,  ps is affirmed, and 
ph is affirmed is excluded. This is evident since  ph cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Axiomax-frege2 38085 If a proposition  ch is a necessary consequence of two propositions  ps and  ph and one of those,  ps, is in turn a necessary consequence of the other, 
ph, then the proposition  ch is a necessary consequence of the latter one,  ph, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theoremrp-simp2-frege 38086 Simplification of triple conjunction. Compare with simp2 1062. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ps )
 ) )
 
Theoremrp-simp2 38087 Simplification of triple conjunction. Identical to simp2 1062. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ps )
 
Theoremrp-frege3g 38088 Add antecedent to ax-frege2 38085. More general statement than frege3 38089. Like ax-frege2 38085, it is essentially a closed form of mpd 15, however it has an extra antecedent.

It would be more natural to prove from a1i 11 and ax-frege2 38085 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

 |-  ( ph  ->  ( ( ps 
 ->  ( ch  ->  th )
 )  ->  ( ( ps  ->  ch )  ->  ( ps  ->  th ) ) ) )
 
Theoremfrege3 38089 Add antecedent to ax-frege2 38085. Special case of rp-frege3g 38088. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ch  ->  (
 ph  ->  ps ) )  ->  ( ( ch  ->  ph )  ->  ( ch  ->  ps ) ) ) )
 
Theoremrp-misc1-frege 38090 Double-use of ax-frege2 38085. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ps ) )  ->  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ch ) ) )
 
Theoremrp-frege24 38091 Introducing an embedded antecedent. Alternate proof for frege24 38109. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ps )  ->  ( ph  ->  ( ch  ->  ps ) ) )
 
Theoremrp-frege4g 38092 Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ps 
 ->  ( ch  ->  th )
 ) )  ->  ( ph  ->  ( ( ps 
 ->  ch )  ->  ( ps  ->  th ) ) ) )
 
Theoremfrege4 38093 Special case of closed form of a2d 29. Special case of rp-frege4g 38092. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( ph  ->  ps )  ->  ( ch  ->  ( ph  ->  ps )
 ) )  ->  (
 ( ph  ->  ps )  ->  ( ( ch  ->  ph )  ->  ( ch  ->  ps ) ) ) )
 
Theoremfrege5 38094 A closed form of syl 17. Identical to imim2 58. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ch  ->  ph )  ->  ( ch  ->  ps ) ) )
 
Theoremrp-7frege 38095 Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( th  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) ) )
 
Theoremrp-4frege 38096 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ( ps  ->  ph )  ->  ch ) )  ->  ( ph  ->  ch ) )
 
Theoremrp-6frege 38097 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  ( ( ps 
 ->  ( ( ch  ->  ps )  ->  th )
 )  ->  ( ps  ->  th ) ) )
 
Theoremrp-8frege 38098 Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ps 
 ->  ( ( ch  ->  ps )  ->  th )
 ) )  ->  ( ph  ->  ( ps  ->  th ) ) )
 
Theoremrp-frege25 38099 Closed form for a1dd 50. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
Theoremfrege6 38100 A closed form of imim2d 57 which is a deduction adding nested antecedents. Proposition 6 of [Frege1879] p. 33. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ph  ->  ( ( th  ->  ps )  ->  ( th  ->  ch ) ) ) )
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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 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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