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Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempfxccatin12lem1 39101 Lemma 1 for pfxccatin12 39103. Could replace swrdccatin12lem2b 12885. (Contributed by AV, 9-May-2020.)
 |-  (
 ( M  e.  (
 0 ... L )  /\  N  e.  ( L ... X ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( K  -  ( L  -  M ) )  e.  ( 0..^ ( N  -  L ) ) ) )
 
Theorempfxccatin12lem2 39102 Lemma 2 for pfxccatin12 39103. Could replace swrdccatin12lem2 12888. (Contributed by AV, 9-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ...
 L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( (
 ( A ++  B ) substr  <. M ,  N >. ) `
  K )  =  ( ( B prefix  ( N  -  L ) ) `
  ( K  -  ( # `  ( A substr  <. M ,  L >. ) ) ) ) ) )
 
Theorempfxccatin12 39103 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 12890. (Contributed by AV, 9-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ...
 L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A ++  B ) substr 
 <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) ++  ( B prefix  ( N  -  L ) ) ) ) )
 
Theorempfxccat3 39104 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 12891. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ...
 N )  /\  N  e.  ( 0 ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A ++  B ) substr 
 <. M ,  N >. )  =  if ( N 
 <_  L ,  ( A substr  <. M ,  N >. ) ,  if ( L 
 <_  M ,  ( B substr  <. ( M  -  L ) ,  ( N  -  L ) >. ) ,  ( ( A substr  <. M ,  L >. ) ++  ( B prefix 
 ( N  -  L ) ) ) ) ) ) )
 
Theorempfxccatpfx1 39105 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V  /\  N  e.  ( 0
 ... L ) ) 
 ->  ( ( A ++  B ) prefix  N )  =  ( A prefix  N ) )
 
Theorempfxccatpfx2 39106 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   &    |-  M  =  ( # `  B )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V  /\  N  e.  ( ( L  +  1 )
 ... ( L  +  M ) ) ) 
 ->  ( ( A ++  B ) prefix  N )  =  ( A ++  ( B prefix  ( N  -  L ) ) ) )
 
Theorempfxccat3a 39107 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. Could replace swrdccat3a 12893. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   &    |-  M  =  ( # `  B )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( N  e.  ( 0 ... ( L  +  M )
 )  ->  ( ( A ++  B ) prefix  N )  =  if ( N 
 <_  L ,  ( A prefix  N ) ,  ( A ++  ( B prefix  ( N  -  L ) ) ) ) ) )
 
Theorempfxccatid 39108 A prefix of a concatenation of length of the first concatenated word is the first word itself. Could replace swrdccatid 12896. (Contributed by AV, 10-May-2020.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A ) )  ->  ( ( A ++  B ) prefix  N )  =  A )
 
Theoremccats1pfxeqbi 39109 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. Could replace ccats1swrdeqbi 12897. (Contributed by AV, 10-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) )  <->  U  =  ( W ++  <" ( lastS  `  U ) "> ) ) )
 
Theorempfxccatin12d 39110 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12d 12900. (Contributed by AV, 10-May-2020.)
 |-  ( ph  ->  ( # `  A )  =  L )   &    |-  ( ph  ->  ( A  e. Word  V 
 /\  B  e. Word  V ) )   &    |-  ( ph  ->  M  e.  ( 0 ...
 L ) )   &    |-  ( ph  ->  N  e.  ( L ... ( L  +  ( # `  B ) ) ) )   =>    |-  ( ph  ->  ( ( A ++  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) ++  ( B prefix  ( N  -  L ) ) ) )
 
Theoremreuccatpfxs1lem 39111* Lemma for reuccatpfxs1 39112. Could replace reuccats1lem 12879. (Contributed by AV, 9-May-2020.)
 |-  (
 ( ( W  e. Word  V 
 /\  U  e.  X )  /\  A. s  e.  V  ( ( W ++ 
 <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  ->  ( W  =  ( U prefix  ( # `  W ) )  ->  U  =  ( W ++  <" S "> ) ) )
 
Theoremreuccatpfxs1 39112* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 12880. (Contributed by AV, 10-May-2020.)
 |-  (
 ( W  e. Word  V  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  ->  ( E! v  e.  V  ( W ++  <" v "> )  e.  X  ->  E! w  e.  X  W  =  ( w prefix  ( # `  W ) ) ) )
 
Theoremsplvalpfx 39113 Value of the substring replacement operator. (Contributed by AV, 11-May-2020.)
 |-  (
 ( S  e.  V  /\  ( F  e.  NN0  /\  T  e.  X  /\  R  e.  Y )
 )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S prefix  F ) ++  R ) ++  ( S substr  <. T ,  ( # `  S ) >. ) ) )
 
Theoremrepswpfx 39114 A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
 |-  (
 ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  ->  ( ( S repeatS  N ) prefix  L )  =  ( S repeatS  L ) )
 
Theoremcshword2 39115 Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( ( W substr  <. ( N 
 mod  ( # `  W ) ) ,  ( # `
  W ) >. ) ++  ( W prefix  ( N  mod  ( # `  W ) ) ) ) )
 
Theorempfxco 39116 Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  A  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  F : A
 --> B )  ->  ( F  o.  ( W prefix  N ) )  =  (
 ( F  o.  W ) prefix  N ) )
 
21.33.7  Auxiliary theorems for graph theory

Additional theorems for classical first-order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory.

 
21.33.7.1  Negated equality and membership - extension
 
Theoremelnelall 39117 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  e.  B  ->  ( A  e/  B  ->  ph ) )
 
21.33.7.2  Subclasses and subsets - extension
 
Theoremclel5 39118* Alternate definition of class membership: a class  X is an element of another class  A iff there is an element of  A equal to  X. (Contributed by AV, 13-Nov-2020.)
 |-  ( X  e.  A  <->  E. x  e.  A  X  =  x )
 
Theoremdfss7 39119* Alternate definition of subclass relationship: a class  A is a subclass of another class  B iff each element of  A is equal to an element of  B. (Contributed by AV, 13-Nov-2020.)
 |-  ( A  C_  B  <->  A. x  e.  A  E. y  e.  B  x  =  y )
 
Theoremsssseq 39120 If a class is a subclass of another class, the classes are equal iff the other class is a subclass of the first class. (Contributed by AV, 23-Dec-2020.)
 |-  ( B  C_  A  ->  ( A  C_  B  <->  A  =  B ) )
 
Theoremprcssprc 39121 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
 |-  (
 ( A  C_  B  /\  A  e/  _V )  ->  B  e/  _V )
 
21.33.7.3  The empty set - extension
 
Theoremralnralall 39122* A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
 |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
 /\  A. x  e.  A  -.  ph )  ->  ps )
 )
 
Theoremfalseral0 39123* A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)
 |-  (
 ( A. x  -.  ph  /\ 
 A. x  e.  A  ph )  ->  A  =  (/) )
 
Theoremralralimp 39124* Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)
 |-  (
 ( ph  /\  A  =/=  (/) )  ->  ( A. x  e.  A  (
 ( ph  ->  ( th  \/  ta ) )  /\  -. 
 th )  ->  ta )
 )
 
Theoremn0rex 39125* There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)
 |-  ( A  =/=  (/)  ->  E. x  e.  A  x  e.  A )
 
Theoremssn0rex 39126* There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)
 |-  (
 ( A  C_  B  /\  A  =/=  (/) )  ->  E. x  e.  B  x  e.  A )
 
21.33.7.4  Unordered and ordered pairs - extension
 
Theoremelpwdifsn 39127 A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.)
 |-  (
 ( S  e.  W  /\  S  C_  V  /\  A  e/  S )  ->  S  e.  ~P ( V  \  { A }
 ) )
 
Theorempr1eqbg 39128 A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =  C 
 <->  { A ,  B }  =  { B ,  C } ) )
 
Theorempr1nebg 39129 A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =/=  C  <->  { A ,  B }  =/=  { B ,  C } ) )
 
Theoremprelpw 39130 A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ( A  e.  C  /\  B  e.  C ) 
 <->  { A ,  B }  e.  ~P C ) )
 
Theoremrexdifpr 39131 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
 |-  ( E. x  e.  ( A  \  { B ,  C } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  x  =/=  C  /\  ph )
 )
 
Theoremissn 39132* A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)
 |-  ( E. x  e.  A  A. y  e.  A  x  =  y  ->  E. z  A  =  { z } )
 
Theoremn0snor2el 39133* A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.)
 |-  ( A  =/=  (/)  ->  ( E. x  e.  A  E. y  e.  A  x  =/=  y  \/  E. z  A  =  { z } )
 )
 
Theoremopidg 39134 The ordered pair  <. A ,  A >. in Kuratowski's representation. Closed form of opid 4199. (Contributed by AV, 18-Sep-2020.)
 |-  ( A  e.  _V  ->  <. A ,  A >.  =  { { A } } )
 
Theoremsnopeqop 39135 Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( { <. A ,  B >. }  =  <. C ,  D >. 
 <->  ( A  =  B  /\  C  =  D  /\  C  =  { A } ) )
 
Theorempropeqop 39136 Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   =>    |-  ( { <. A ,  B >. ,  <. C ,  D >. }  =  <. E ,  F >. 
 <->  ( ( A  =  C  /\  E  =  { A } )  /\  (
 ( A  =  B  /\  F  =  { A ,  D } )  \/  ( A  =  D  /\  F  =  { A ,  B } ) ) ) )
 
Theorempropssopi 39137 If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   =>    |-  ( { <. A ,  B >. ,  <. C ,  D >. }  C_  <. E ,  F >.  ->  A  =  C )
 
Theoremssprss 39138 A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( { A ,  B }  C_  { C ,  D }  <->  ( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D )
 ) ) )
 
Theoremssprsseq 39139 A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  ( { A ,  B }  C_  { C ,  D }  <->  { A ,  B }  =  { C ,  D } ) )
 
Theoremelpr2elpr 39140* For an element of an unordered pair which is a subset of a given set, there is another (maybe the same) element of the given set being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)
 |-  (
 ( X  e.  V  /\  Y  e.  V  /\  A  e.  { X ,  Y } )  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b } )
 
21.33.7.5  Indexed union and intersection - extension
 
Theoremiunopeqop 39141* Equivalence for an ordered pair equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.)
 |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  (/)  ->  ( U_ x  e.  A  { <. x ,  B >. }  =  <. C ,  D >.  ->  E. z  A  =  { z } )
 )
 
TheoremotiunsndisjX 39142* The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  ( B  e.  X  -> Disj  a  e.  V  U_ c  e.  W  { <. a ,  B ,  c >. } )
 
21.33.7.6  Ordered-pair class abstractions - extension
 
Theoremopabn1stprc 39143* An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wwf. (Contributed by AV, 27-Dec-2020.)
 |-  ( E. y ph  ->  { <. x ,  y >.  |  ph } 
 e/  _V )
 
21.33.7.7  Introduce the Axiom of Power Sets - extension
 
Theoremralxfrd2 39144* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. Variant of ralxfrd 4631. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  y  e.  C  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  ch ) )
 
Theoremrexxfrd2 39145* Transfer existence from a variable 
x to another variable  y contained in expression  A. Variant of rexxfrd 4632. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  y  e.  C  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  B  ps 
 <-> 
 E. y  e.  C  ch ) )
 
21.33.7.8  Relations - extension
 
Theoremresresdm 39146 A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.)
 |-  ( F  =  ( E  |`  A )  ->  F  =  ( E  |`  dom  F ) )
 
Theoremresisresindm 39147 The restriction of a relation by a set  B is identical with the restriction by the intersection of  B with the domain of the relation. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  ( Rel  F  ->  ( F  |`  B )  =  ( F  |`  ( B  i^i  dom  F ) ) )
 
Theoremssrelrn 39148* If a relation is a subset of a cartesian product, then for each element of the range of the relation there is an element of the first set of the cartesian product which is related to the element of the range by the relation. (Contributed by AV, 24-Oct-2020.)
 |-  (
 ( R  C_  ( A  X.  B )  /\  Y  e.  ran  R ) 
 ->  E. a  e.  A  a R Y )
 
21.33.7.9  Functions - extension
 
Theoremfvifeq 39149 Equality of function values with conditional arguments, see also fvif 5903. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  ( A  =  if ( ph ,  B ,  C )  ->  ( F `  A )  =  if ( ph ,  ( F `
  B ) ,  ( F `  C ) ) )
 
Theorem2f1fvneq 39150 If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  (
 ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C )  /\  A  =/=  B ) 
 ->  ( ( ( E `
  ( F `  A ) )  =  X  /\  ( E `
  ( F `  B ) )  =  Y )  ->  X  =/=  Y ) )
 
Theoremf1cofveqaeq 39151 If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021.)
 |-  (
 ( ( F : B -1-1-> C  /\  G : A -1-1-> B )  /\  ( X  e.  A  /\  Y  e.  A )
 )  ->  ( ( F `  ( G `  X ) )  =  ( F `  ( G `  Y ) ) 
 ->  X  =  Y ) )
 
Theoremf1cofveqaeqALT 39152 Alternate proof of f1cofveqaeq 39151, 1 essential step shorter, but having more bytes (305 vs. 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( ( F : B -1-1-> C  /\  G : A -1-1-> B )  /\  ( X  e.  A  /\  Y  e.  A )
 )  ->  ( ( F `  ( G `  X ) )  =  ( F `  ( G `  Y ) ) 
 ->  X  =  Y ) )
 
Theoremrnfdmpr 39153 The range of a one-to-one function 
F of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  (
 ( X  e.  V  /\  Y  e.  W ) 
 ->  ( F  Fn  { X ,  Y }  ->  ran  F  =  {
 ( F `  X ) ,  ( F `  Y ) } )
 )
 
Theoremimarnf1pr 39154 The image of the range of a function  F under a function  E if  F is a function of a pair into the domain of  E. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  (
 ( X  e.  V  /\  Y  e.  W ) 
 ->  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R ) 
 /\  ( ( E `
  ( F `  X ) )  =  A  /\  ( E `
  ( F `  Y ) )  =  B ) )  ->  ( E " ran  F )  =  { A ,  B } ) )
 
Theoremfuniun 39155* A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.)
 |-  ( Fun  F  ->  F  =  U_ x  e.  dom  F { <. x ,  ( F `  x ) >. } )
 
Theoremfunopsn 39156* If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  (
 ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  E. a
 ( X  =  {
 a }  /\  F  =  { <. a ,  a >. } ) )
 
Theoremfunop 39157* An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  ( Fun  <. X ,  Y >.  <->  E. a ( X  =  { a }  /\  <. X ,  Y >.  =  { <. a ,  a >. } ) )
 
Theoremfunop1 39158* A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
 |-  ( E. x E. y  F  =  <. x ,  y >.  ->  ( Fun  F  <->  E. x E. y  F  =  { <. x ,  y >. } ) )
 
Theoremf1ssf1 39159 A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
 |-  (
 ( Fun  F  /\  Fun  `' F  /\  G  C_  F )  ->  Fun  `' G )
 
Theoremfunsndifnop 39160 A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.)
 |-  A  e.  V   &    |-  B  e.  W   &    |-  G  =  { <. A ,  B >. }   =>    |-  ( A  =/=  B  ->  -.  G  e.  ( _V  X.  _V ) )
 
Theoremfunsneqopsn 39161 A singleton of an ordered pair is an ordered pair of equal singletons if the components are equal. (Contributed by AV, 24-Sep-2020.)
 |-  A  e.  V   &    |-  B  e.  W   &    |-  G  =  { <. A ,  B >. }   =>    |-  ( A  =  B  ->  G  =  <. { A } ,  { A } >. )
 
Theoremfunsneqop 39162 A singleton of an ordered pair is an ordered pair if the components are equal. (Contributed by AV, 24-Sep-2020.)
 |-  A  e.  V   &    |-  B  e.  W   &    |-  G  =  { <. A ,  B >. }   =>    |-  ( A  =  B  ->  G  e.  ( _V 
 X.  _V ) )
 
Theoremfunsneqopb 39163 A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.)
 |-  A  e.  V   &    |-  B  e.  W   &    |-  G  =  { <. A ,  B >. }   =>    |-  ( A  =  B  <->  G  e.  ( _V  X.  _V ) )
 
Theoremfundmge2nop 39164 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.)
 |-  (
 ( Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  -.  G  e.  ( _V  X.  _V ) )
 
Theoremfun2dmnop 39165 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 12-Oct-2020.)
 |-  A  e.  V   &    |-  B  e.  W   =>    |-  (
 ( Fun  G  /\  A  =/=  B  /\  { A ,  B }  C_ 
 dom  G )  ->  -.  G  e.  ( _V  X.  _V ) )
 
Theoremfun2dmnopgexmpl 39166 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.)
 |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  -.  G  e.  ( _V 
 X.  _V ) )
 
Theoremopabresex0d 39167* A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.)
 |-  (
 ( ph  /\  x R y )  ->  x  e.  C )   &    |-  ( ( ph  /\  x R y ) 
 ->  th )   &    |-  ( ( ph  /\  x  e.  C ) 
 ->  { y  |  th }  e.  V )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  { <. x ,  y >.  |  ( x R y  /\  ps ) }  e.  _V )
 
Theoremopabbrfex0d 39168* A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.)
 |-  (
 ( ph  /\  x R y )  ->  x  e.  C )   &    |-  ( ( ph  /\  x R y ) 
 ->  th )   &    |-  ( ( ph  /\  x  e.  C ) 
 ->  { y  |  th }  e.  V )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  { <. x ,  y >.  |  x R y }  e.  _V )
 
Theoremopabresexd 39169* A collection of ordered pairs, the second component being a function, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
 |-  (
 ( ph  /\  x R y )  ->  x  e.  C )   &    |-  ( ( ph  /\  x R y ) 
 ->  y : A --> B )   &    |-  ( ( ph  /\  x  e.  C )  ->  A  e.  U )   &    |-  ( ( ph  /\  x  e.  C ) 
 ->  B  e.  V )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  { <. x ,  y >.  |  ( x R y  /\  ps ) }  e.  _V )
 
Theoremopabbrfexd 39170* A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.)
 |-  (
 ( ph  /\  x R y )  ->  x  e.  C )   &    |-  ( ( ph  /\  x R y ) 
 ->  y : A --> B )   &    |-  ( ( ph  /\  x  e.  C )  ->  A  e.  U )   &    |-  ( ( ph  /\  x  e.  C ) 
 ->  B  e.  V )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  { <. x ,  y >.  |  x R y }  e.  _V )
 
Theoremopabresex2d 39171* Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
 |-  (
 ( ph  /\  x ( W `  G ) y )  ->  ps )   &    |-  ( ph  ->  { <. x ,  y >.  |  ps }  e.  V )   =>    |-  ( ph  ->  { <. x ,  y >.  |  ( x ( W `  G ) y  /\  th ) }  e.  _V )
 
Theoremmptmpt2opabbrd 39172* The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.)
 |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  X  e.  ( A `  G ) )   &    |-  ( ph  ->  Y  e.  ( B `  G ) )   &    |-  ( ph  ->  { <. f ,  h >.  |  ps }  e.  V )   &    |-  (
 ( ph  /\  f ( D `  G ) h )  ->  ps )   &    |-  (
 ( a  =  X  /\  b  =  Y )  ->  ( ta  <->  th ) )   &    |-  (
 g  =  G  ->  ( ch  <->  ta ) )   &    |-  M  =  ( g  e.  _V  |->  ( a  e.  ( A `  g ) ,  b  e.  ( B `
  g )  |->  {
 <. f ,  h >.  |  ( ch  /\  f
 ( D `  g
 ) h ) }
 ) )   =>    |-  ( ph  ->  ( X ( M `  G ) Y )  =  { <. f ,  h >.  |  ( th  /\  f ( D `
  G ) h ) } )
 
Theoremmptmpt2opabovd 39173* The operation value of a function value of a collection of ordered pairs of related elements (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.)
 |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  X  e.  ( A `  G ) )   &    |-  ( ph  ->  Y  e.  ( B `  G ) )   &    |-  ( ph  ->  { <. f ,  h >.  |  ps }  e.  V )   &    |-  (
 ( ph  /\  f ( D `  G ) h )  ->  ps )   &    |-  M  =  ( g  e.  _V  |->  ( a  e.  ( A `  g ) ,  b  e.  ( B `
  g )  |->  {
 <. f ,  h >.  |  ( f ( a ( C `  g
 ) b ) h 
 /\  f ( D `
  g ) h ) } ) )   =>    |-  ( ph  ->  ( X ( M `  G ) Y )  =  { <. f ,  h >.  |  ( f ( X ( C `  G ) Y ) h  /\  f ( D `  G ) h ) } )
 
Theoremfpropnf1 39174 A function, given by an unordered pair of ordered pairs, which is not injective/one-to-one. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
 |-  F  =  { <. X ,  Z >. ,  <. Y ,  Z >. }   =>    |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  /\  X  =/=  Y )  ->  ( Fun  F 
 /\  -.  Fun  `' F ) )
 
21.33.7.10  Restricted iota - extension
 
Theoremriotaeqimp 39175* If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)
 |-  I  =  ( iota_ a  e.  V  X  =  A )   &    |-  J  =  ( iota_ a  e.  V  Y  =  A )   &    |-  ( ph  ->  E! a  e.  V  X  =  A )   &    |-  ( ph  ->  E! a  e.  V  Y  =  A )   =>    |-  ( ( ph  /\  I  =  J )  ->  X  =  Y )
 
21.33.7.11  Equinumerosity - extension
 
Theoremresfnfinfin 39176 The restriction of a function by a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  (
 ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  B )  e.  Fin )
 
Theoremresidfi 39177 A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
 |-  (
 (  _I  |`  A )  e.  Fin  <->  A  e.  Fin )
 
21.33.7.12  Subtraction - extension
 
Theoremcnambpcma 39178 ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  -  B )  +  C )  -  A )  =  ( C  -  B ) )
 
Theoremcnapbmcpd 39179 ((a+b)-c)+d = ((a+d)+b)-c holds for complex numbers a,b,c,d. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( ( A  +  B )  -  C )  +  D )  =  (
 ( ( A  +  D )  +  B )  -  C ) )
 
21.33.7.13  Multiplication - extension
 
Theorem2txmxeqx 39180 Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( X  e.  CC  ->  ( ( 2  x.  X )  -  X )  =  X )
 
21.33.7.14  Ordering on reals (cont.) - extension
 
Theoremleaddsuble 39181 Addition and subtraction on one side of 'less or equal'. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  ( ( A  +  B )  -  C )  <_  A ) )
 
Theorem2leaddle2 39182 If two real numbers are less than a third real number, the sum of the real numbers is less than twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  C 
 /\  B  <  C )  ->  ( A  +  B )  <  ( 2  x.  C ) ) )
 
Theoremltnltne 39183 Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( -.  B  <  A  /\  -.  B  =  A ) ) )
 
Theoremp1lep2 39184 A real number increasd by 1 is less than or equal to the number increased by 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( N  e.  RR  ->  ( N  +  1 ) 
 <_  ( N  +  2 ) )
 
Theoremlelttrdi 39185 If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
 |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )
 )   &    |-  ( ph  ->  B  <_  C )   =>    |-  ( ph  ->  ( A  <  B  ->  A  <  C ) )
 
Theoremltsubsubaddltsub 39186 If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
 |-  (
 ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
 )  ->  ( J  <  ( ( L  -  M )  -  N ) 
 <->  ( J  +  M )  <  ( L  -  N ) ) )
 
Theoremzm1nn 39187 An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
 |-  (
 ( N  e.  NN0  /\  L  e.  ZZ )  ->  ( ( J  e.  RR  /\  0  <_  J  /\  J  <  ( ( L  -  N )  -  1 ) ) 
 ->  ( L  -  1
 )  e.  NN )
 )
 
21.33.7.15  Nonnegative integers (as a subset of complex numbers) - extension
 
Theoremlesubnn0 39188 Subtracting a nonnegative integer from a nonnegative integer which is greater than or equal to the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( B  <_  A  ->  ( A  -  B )  e.  NN0 ) )
 
Theoremltsubnn0 39189 Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( B  <  A  ->  ( A  -  B )  e.  NN0 ) )
 
Theoremnn0resubcl 39190 Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  -  B )  e.  RR )
 
21.33.7.16  Upper sets of integers - extension
 
Theoremeluzge0nn0 39191 If an integer is greater than or equal to a nonnegative integer, then it is a nonnegative integer. (Contributed by Alexander van der Vekens, 27-Aug-2018.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( 0 
 <_  M  ->  N  e.  NN0 ) )
 
21.33.7.17  Finite intervals of integers - extension
 
Theoremssfz12 39192 Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( K  e.  ZZ  /\  L  e.  ZZ  /\  K  <_  L )  ->  ( ( K ... L )  C_  ( M ... N )  ->  ( M  <_  K  /\  L  <_  N ) ) )
 
Theoremelfz2z 39193 Membership of an integer in a finite set of sequential integers starting at 0. (Contributed by Alexander van der Vekens, 25-May-2018.)
 |-  (
 ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  (
 0 ... N )  <->  ( 0  <_  K  /\  K  <_  N ) ) )
 
Theorem2elfz3nn0 39194 If there are two elements in a finite set of sequential integers starting at 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( A  e.  NN0  /\  B  e.  NN0  /\  N  e.  NN0 ) )
 
Theoremfz0addcom 39195 The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( A  +  B )  =  ( B  +  A ) )
 
Theorem2elfz2melfz 39196 If the sum of two integers of a 0 based finite set of sequential integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the 0 based finite set of sequential integers with the first integer as upper bound. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( N  <  ( A  +  B )  ->  ( B  -  ( N  -  A ) )  e.  ( 0 ...
 A ) ) )
 
Theoremfz0addge0 39197 The sum of two integers in 0 based finite sets of sequential integers is greater than or equal to zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  (
 ( A  e.  (
 0 ... M )  /\  B  e.  ( 0 ... N ) )  -> 
 0  <_  ( A  +  B ) )
 
Theoremelfzlble 39198 Membership of an integer in a finite set of sequential integers with the integer as upper bound and a lower bound less than or equal to the integer. (Contributed by AV, 21-Oct-2018.)
 |-  (
 ( N  e.  ZZ  /\  M  e.  NN0 )  ->  N  e.  ( ( N  -  M )
 ... N ) )
 
Theoremelfzelfzlble 39199 Membership of an element of a finite set of sequential integers in a finite set of sequential integers with the same upper bound and a lower bound less than the upper bound. (Contributed by AV, 21-Oct-2018.)
 |-  (
 ( M  e.  ZZ  /\  K  e.  ( 0
 ... N )  /\  N  <  ( M  +  K ) )  ->  K  e.  ( ( N  -  M ) ... N ) )
 
21.33.7.18  Half-open integer ranges - extension
 
Theoremsubsubelfzo0 39200 Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  (
 ( A  e.  (
 0..^ N )  /\  I  e.  ( 0..^ N )  /\  -.  I  <  ( N  -  A ) )  ->  ( I  -  ( N  -  A ) )  e.  ( 0..^ A ) )
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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 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 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-40909
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