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Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiunrelexp0 36201* Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
 |-  (
 ( R  e.  V  /\  Z  C_  NN0  /\  ( { 0 ,  1 }  i^i  Z )  =/=  (/) )  ->  ( U_ x  e.  Z  ( R ^r  x ) ^r 
 0 )  =  ( R ^r  0 ) )
 
Theoremrelexpxpnnidm 36202 Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
 |-  ( N  e.  NN  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  (
 ( A  X.  B ) ^r  N )  =  ( A  X.  B ) ) )
 
Theoremrelexpiidm 36203 Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
 |-  (
 ( A  e.  V  /\  N  e.  NN0 )  ->  ( (  _I  |`  A ) ^r  N )  =  (  _I  |`  A ) )
 
Theoremrelexpss1d 36204 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A ^r  N ) 
 C_  ( B ^r  N ) )
 
Theoremcomptiunov2i 36205* The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)
 |-  X  =  ( a  e.  _V  |->  U_ i  e.  I  ( a  .^  i )
 )   &    |-  Y  =  ( b  e.  _V  |->  U_ j  e.  J  ( b  .^  j ) )   &    |-  Z  =  ( c  e.  _V  |->  U_ k  e.  K  ( c  .^  k )
 )   &    |-  I  e.  _V   &    |-  J  e.  _V   &    |-  K  =  ( I  u.  J )   &    |-  U_ k  e.  I  ( d  .^  k )  C_  U_ i  e.  I  ( U_ j  e.  J  ( d  .^  j ) 
 .^  i )   &    |-  U_ k  e.  J  ( d  .^  k )  C_  U_ i  e.  I  ( U_ j  e.  J  (
 d  .^  j )  .^  i )   &    |-  U_ i  e.  I  ( U_ j  e.  J  ( d  .^  j ) 
 .^  i )  C_  U_ k  e.  ( I  u.  J ) ( d  .^  k )   =>    |-  ( X  o.  Y )  =  Z
 
Theoremcorclrcl 36206 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 r*  o.  r* )  =  r*
 
Theoremiunrelexpmin1 36207* The indexed union of relation exponentiation over the natural numbers is the minimum 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  =  NN )  ->  A. s ( ( R  C_  s  /\  ( s  o.  s
 )  C_  s )  ->  ( C `  R )  C_  s ) )
 
Theoremrelexpmulnn 36208 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 36209 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 36210* 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 36211* 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 36212 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 36213 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 36214 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 36215 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 36216 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 36217 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 13054 or when the sum of the powers isn't 1 as shown by relexpaddg 13053. (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 36218* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 13060. (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 ) )
 
21.25.2.4  Transitive closure of a relation
 
Theoremdftrcl3 36219* 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 36220* 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 36221 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 36222 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 36223 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 36224 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
 |-  (
 t+  o.  t+ )  =  t+
 
Theoremtrclimalb2 36225 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 36226* 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 36227 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 36228 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 36229 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 36230 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 36231 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 36232* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 13062. (Contributed by RP, 5-Jun-2020.)
 |-  t*  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r  n ) )
 
Theorembrfvrtrcld 36233* 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 13057. (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 36234 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 36235 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 36236 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 36237 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 36238 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
 |-  (
 r*  o.  t+ )  =  t*
 
Theoremcortrcltrcl 36239 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  t+ )  =  t*
 
Theoremcorclrtrcl 36240 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 r*  o.  t* )  =  t*
 
Theoremcotrclrcl 36241 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
 |-  (
 t+  o.  r* )  =  t*
 
Theoremcortrclrcl 36242 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  r* )  =  t*
 
Theoremcotrclrtrcl 36243 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t+  o.  t* )  =  t*
 
Theoremcortrclrtrcl 36244 The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
 |-  (
 t*  o.  t* )  =  t*
 
21.25.2.5  Adapted from Frege

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

 
Theoremfrege77d 36245 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 36443. (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 36246 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 36447. (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 36247 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 36449. (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 36248 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 36462. (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 36249 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 36453. (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 36250 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 36457. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  A ( t+ `  R ) B )
 
Theoremfrege97d 36251 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 36463. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  =  ( ( t+ `
  R ) " U ) )   =>    |-  ( ph  ->  ( R " A ) 
 C_  A )
 
Theoremfrege98d 36252 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 36464. (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 36253 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 36468. (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 36254 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 36472. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( A R B  \/  A  =  B ) )
 
Theoremfrege108d 36255 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 36474. (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 36256 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 36475. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  R  e.  _V )   &    |-  ( ph  ->  A  =  ( U  u.  (
 ( t+ `  R ) " U ) ) )   =>    |-  ( ph  ->  ( R " A ) 
 C_  A )
 
Theoremfrege114d 36257 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 36480. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  ( A R B  \/  A  =  B ) )   =>    |-  ( ph  ->  ( A R B  \/  A  =  B  \/  B R A ) )
 
Theoremfrege111d 36258 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 36477. (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 36259 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 36488. (Contributed by RP, 15-Jul-2020.)
 |-  ( ph  ->  A  =  ( F `  X ) )   &    |-  ( ph  ->  B  =  ( F `  X ) )   =>    |-  ( ph  ->  ( A ( t+ `
  F ) B  \/  A  =  B ) )
 
Theoremfrege124d 36260 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 36490. (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 36261 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 36492. (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 36262 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 36495. (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 36263 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 36497. (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 36264 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 36499. (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 ) )
 
21.25.3  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 3234 for discussion of an example of a class that is not a set.

Numbered propositions from [Frege1879]. ax-frege1 36293, ax-frege2 36294, ax-frege8 36312, ax-frege28 36333, ax-frege31 36337, ax-frege41 36348, frege52 (see ax-frege52a 36360, frege52b 36392, and ax-frege52c 36391 for translations), frege54 (see ax-frege54a 36365, frege54b 36396 and ax-frege54c 36395 for translations) and frege58 (see ax-frege58a 36378, ax-frege58b 36404 and frege58c 36424 for translations) are considered "core" or axioms. However, at least ax-frege8 36312 can be derived from ax-frege1 36293 and ax-frege2 36294, see axfrege8 36310.

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 36360, frege52b 36392, and ax-frege52c 36391. In dffrege69 36435, 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 36275 for a definition in terms of image and subset. In dffrege76 36442, 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 36465, 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 36481, 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 36245 for an example.

 
21.25.3.1  _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 371, df-an 372, dfxor4 36265, dfxor5 36266.

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 1914 (or simpl 458 and simpr 462 and anifp 1428 in the propositional case) and statments similar to cbvalivw 1842, ax-gen 1663, alrimiv 1767, and alrimdv 1769. 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 1692, 
A  =  _V eqv 3714; Some are not B,  -.  A. x x  e.  B,  E. x -.  x  e.  B exnal 1693, 
B  C.  _V pssv 3770,  B  =/=  _V nev 36269; There are no C,  A. x -.  x  e.  C,  -.  E. x x  e.  C alnex 1659, 
C  =  (/) eq0 3713; There exist D,  -. 
A. x -.  x  e.  D,  E. x x  e.  D df-ex 1658,  (/)  C.  D 0pss 3768,  D  =/=  (/) n0 3707.

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 36270, 
E  =  ( E  i^i  P ) df-ss 3386,  E  C_  P dfss2 3389; No F are P,  A. x ( x  e.  F  ->  -.  x  e.  P ),  -.  E. x
( x  e.  F  /\  x  e.  P
) alinexa 1707,  ( F  i^i  P
)  =  (/) disj1 3773; Some G are not P,  -.  A. x ( x  e.  G  ->  x  e.  P ),  E. x ( x  e.  G  /\  -.  x  e.  P
) exanali 1715,  ( G  i^i  P
)  C.  G nssinpss 3641,  -.  G  C_  P nss 3458; Some H are P,  -.  A. x
( x  e.  H  ->  -.  x  e.  P
),  E. x ( x  e.  H  /\  x  e.  P ) bj-exnalimn 31157,  (/)  C.  ( H  i^i  P
) 0pssin 36272, 
( H  i^i  P
)  =/=  (/) ndisj 36271.

 
Theoremdfxor4 36265 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 36266 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 36267 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 36268 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 36269* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  =/=  _V  <->  -.  A. x  x  e.  A )
 
Theoremdfss6 36270* Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  C_  B  <->  -.  E. x ( x  e.  A  /\  -.  x  e.  B ) )
 
Theoremndisj 36271* 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 36272* 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 ) )
 
21.25.3.2  _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 wasn't 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 36273 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 36274 The property of relation  R being hereditary in class  A.
 wff  R hereditary  A
 
Definitiondf-he 36275 The property of relation  R being hereditary in class  A. (Contributed by RP, 27-Mar-2020.)
 |-  ( R hereditary  A  <->  ( R " A )  C_  A )
 
Theoremdfhe2 36276 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 36277* 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 36278 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 36279 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( R  =  S  ->  ( R hereditary  A  <->  S hereditary  A ) )
 
Theoremheeq2 36280 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( A  =  B  ->  ( R hereditary  A  <->  R hereditary  B ) )
 
Theoremsbcheg 36281 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 36282 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 36283 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
 
TheoremxpheOLD 36284 Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) Obsolete version of xphe 36283 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  X.  B ) hereditary  B
 
Theorem0he 36285 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
 |-  (/) hereditary  A
 
Theorem0heALT 36286 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 36287 Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
 |-  A hereditary  (/)
 
Theoremunhe1 36288 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 36289 Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
 |-  { <. A ,  A >. } hereditary  { B }
 
Theoremidhe 36290 The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
 |-  _I hereditary  A
 
Theorempsshepw 36291 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 36292 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
 
21.25.3.3  _Begriffsschrift_ Chapter II Implication
 
Axiomax-frege1 36293 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 36294 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 36295 Simplification of triple conjunction. Compare with simp2 1006. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ps )
 ) )
 
Theoremrp-simp2 36296 Simplification of triple conjunction. Identical to simp2 1006. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ps )
 
Theoremrp-frege3g 36297 Add antecedent to ax-frege2 36294. More general statement than frege3 36298. Like ax-frege2 36294, 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 36294 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

 |-  ( ph  ->  ( ( ps 
 ->  ( ch  ->  th )
 )  ->  ( ( ps  ->  ch )  ->  ( ps  ->  th ) ) ) )
 
Theoremfrege3 36298 Add antecedent to ax-frege2 36294. Special case of rp-frege3g 36297. 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 36299 Double-use of ax-frege2 36294. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ps ) )  ->  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ch ) ) )
 
Theoremrp-frege24 36300 Introducing an embedded antecedent. Alternate proof for frege24 36318. Closed form for a1d 26. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ps )  ->  ( ph  ->  ( ch  ->  ps ) ) )
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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 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40127
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