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Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfzelfzlble 32601 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 then the upper bound. (Contributed by AV, 21-Oct-2018.)
 |-  (
 ( M  e.  ZZ  /\  K  e.  ( 0
 ... N )  /\  N  <  ( M  +  K ) )  ->  K  e.  ( ( N  -  M ) ... N ) )
 
21.25.4.16  Half-open integer ranges - extension
 
Theoremsubsubelfzo0 32602 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 ) )
 
Theoremel2fzo 32603 The lower limit of a half-open integer range which is equal to a nonempty half-open integer range is element of the half-open integer range. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ( ( M..^ N )  =  ( J..^ K )  ->  J  e.  ( J..^ K ) ) )
 
Theoremfzoopth 32604 A half-open integer range can represent an ordered pair, analogous to fzopth 11746. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ( ( M..^ N )  =  ( J..^ K )  <->  ( M  =  J  /\  N  =  K ) ) )
 
Theorem2ffzoeq 32605* Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N )
 --> Y ) )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `
  i )  =  ( P `  i
 ) ) ) )
 
Theoremfzosplitpr 32606 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  ( B  e.  ( ZZ>= `  A )  ->  ( A..^ ( B  +  2 ) )  =  ( ( A..^ B )  u.  { B ,  ( B  +  1
 ) } ) )
 
21.25.4.17  Finite and infinite sums - extension
 
Theoremfsummsndifre 32607* A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\ 
 A. k  e.  A  B  e.  ZZ )  -> 
 sum_ k  e.  ( A  \  { X }
 ) B  e.  RR )
 
Theoremfsumsplitsndif 32608* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\  X  e.  A  /\  A. k  e.  A  B  e.  ZZ )  ->  sum_ k  e.  A  B  =  (
 sum_ k  e.  ( A  \  { X }
 ) B  +  [_ X  /  k ]_ B ) )
 
Theoremfsummmodsndifre 32609* A finite sum of summands modulo a positive number with one of its summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\  N  e.  NN  /\  A. k  e.  A  B  e.  ZZ )  ->  sum_ k  e.  ( A  \  { X } ) ( B 
 mod  N )  e.  RR )
 
Theoremfsummmodsnunz 32610* A finite sum of summands modulo a positive number with an additional summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
 |-  (
 ( A  e.  Fin  /\  N  e.  NN  /\  A. k  e.  ( A  u.  { z }
 ) B  e.  ZZ )  ->  sum_ k  e.  ( A  u.  { z }
 ) ( B  mod  N )  e.  ZZ )
 
21.25.5  Graph theory
 
21.25.5.1  Undirected hypergraphs
 
Theoremuhgraedgrnv 32611 An edge of an undirected hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  (
 ( V UHGrph  E  /\  F  e.  ran  E  /\  N  e.  F )  ->  N  e.  V )
 
21.25.5.2  Walks, Paths and Cycles
 
Theoremwlkc 32612* A walk as class. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
 |-  ( W  e.  ( V Walks  E )  ->  E. f E. p  f ( V Walks  E ) p )
 
Theoremusgra2pthspth 32613 In a undirected simple graph, any path of length 2 is a simple path. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  (
 ( V USGrph  E  /\  ( # `  F )  =  2 )  ->  ( F ( V Paths  E ) P  <->  F ( V SPaths  E ) P ) )
 
Theoremspthdifv 32614 The vertices of a simple path are distinct, so the vertex function is one-to-one. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( F ( V SPaths  E ) P  ->  P :
 ( 0 ... ( # `
  F ) )
 -1-1-> V )
 
Theoremusgra2pthlem1 32615* Lemma for usgra2pth 32616. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  (
 ( F : ( 0..^ ( # `  F ) ) -1-1-> dom  E  /\  P : ( 0
 ... ( # `  F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }
 )  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2 )  ->  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y }
 ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  y  /\  ( P `
  2 )  =  z )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  y }  /\  ( E `
  ( F `  1 ) )  =  { y ,  z } ) ) ) )
 
Theoremusgra2pth 32616* In a undirected simply graph, there is a path of length 2 if and only if there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
 ( 0..^ 2 )
 -1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `  0
 )  =  x  /\  ( P `  1 )  =  y  /\  ( P `  2 )  =  z )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  y }  /\  ( E `
  ( F `  1 ) )  =  { y ,  z } ) ) ) ) )
 
Theoremusgra2pth0 32617* In a undirected simply graph, there is a path of length 2 if and only if there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
 ( 0..^ 2 )
 -1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `  0
 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `
  ( F `  1 ) )  =  { z ,  y } ) ) ) ) )
 
Theoremusgra2adedglem1 32618 In an undirected simple graph, two adjacent edges are an unordered pair of unordered pairs. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. }   &    |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 ->  { { A ,  B } ,  { B ,  C } }  =  ( E " ran  F ) ) )
 
21.25.5.3  Vertex degree (extension)
 
Theoremvdusgravaledg 32619* The value of the vertex degree function for simple undirected graphs in terms of edges. (Contributed by Alexander van der Vekens, 9-Jul-2018.)
 |-  (
 ( V USGrph  E  /\  U  e.  V )  ->  ( ( V VDeg  E ) `  U )  =  ( # `  { x  e.  V  |  { U ,  x }  e.  ran  E } ) )
 
Theoremusgrauvtxvd 32620 In a finite complete undirected simple graph with n vertices every vertex has degree (n-1). (Contributed by Alexander van der Vekens, 9-Jul-2018.)
 |-  (
 ( V USGrph  E  /\  V  e.  Fin )  ->  ( K  e.  ( V UnivVertex  E )  ->  (
 ( V VDeg  E ) `  K )  =  ( ( # `  V )  -  1 ) ) )
 
Theoremvdcusgra 32621* In a finite complete undirected simple graph with n vertices every vertex has degree (n-1). (Contributed by Alexander van der Vekens, 9-Jul-2018.)
 |-  (
 ( V ComplUSGrph  E  /\  V  e.  Fin )  ->  A. v  e.  V  ( ( V VDeg 
 E ) `  v
 )  =  ( ( # `  V )  -  1 ) )
 
21.25.5.4  Undirected hypergraphs as extensible structures
 
Syntaxcedgf 32622 Extend class notation with an edge function.
 class .ef
 
Syntaxcuhgr 32623 Extend class notation with undirected hypergraphs as extensible structures.
 class UHGraph
 
Syntaxcushgr 32624 Extend class notation with undirected simple hypergraphs as extensible structures.
 class USHGraph
 
Definitiondf-edgf 32625 Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)
 |- .ef  = Slot ; 1 6
 
Definitiondf-uhgr 32626* Define the class of all undirected hypergraphs. An undirected hypergraph is a set, regarded as set of "vertices", and a function into the powerset of this set (the empty set excluded), regarded as indexed "edges" connecting vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |- UHGraph  =  {
 g  |  [. ( Base `  g )  /  v ]. [. ( .ef  `  g )  /  e ]. e : dom  e --> ( ~P v  \  { (/)
 } ) }
 
Definitiondf-ushgr 32627* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph  <. V ,  E >. where  E is an injective (one-to-one) function into subsets of  V, representing the (one or more) vertices incident to the edge. This definition corresponds to definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subsets of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E a (non-empty) subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.)
 |- USHGraph  =  {
 g  |  [. ( Base `  g )  /  v ]. [. ( .ef  `  g )  /  e ]. e : dom  e -1-1-> ( ~P v  \  { (/)
 } ) }
 
Theoremisuhgr 32628 The predicate "is an undirected hypergraph." (Contributed by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  ( .ef  `  G )   =>    |-  ( G  e.  U  ->  ( G  e. UHGraph  <->  E : dom  E --> ( ~P V  \  { (/)
 } ) ) )
 
Theoremisushgr 32629 The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  ( .ef  `  G )   =>    |-  ( G  e.  U  ->  ( G  e. USHGraph  <->  E : dom  E -1-1-> ( ~P V  \  { (/)
 } ) ) )
 
Theoremuhgf 32630 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  ( .ef  `  G )   =>    |-  ( G  e. UHGraph  ->  E : dom  E --> ( ~P V  \  { (/) } )
 )
 
Theoremuhgss 32631 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  ( .ef  `  G )   =>    |-  ( ( G  e. UHGraph  /\  F  e.  dom  E )  ->  ( E `  F )  C_  V )
 
Theoremuhgeq12g 32632 If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  E  =  ( .ef  `  G )   &    |-  W  =  ( Base `  H )   &    |-  F  =  ( .ef  `  H )   =>    |-  ( ( ( G  e.  X  /\  H  e.  Y )  /\  ( V  =  W  /\  E  =  F )
 )  ->  ( G  e. UHGraph  <->  H  e. UHGraph  ) )
 
Theoremushguhg 32633 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.)
 |-  ( G  e. USHGraph  ->  G  e. UHGraph  )
 
Theoremuhguhgra 32634 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.)
 |-  (
 ( G  e. UHGraph  /\  V  =  ( Base `  G )  /\  E  =  ( .ef  `  G ) )  ->  V UHGrph  E )
 
Theoremuhgrauhg 32635 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.)
 |-  (
 ( V UHGrph  E  /\  V  =  ( Base `  G )  /\  E  =  ( .ef  `  G ) )  ->  ( G  e.  W  ->  G  e. UHGraph  ) )
 
Theoremuhgrauhgbi 32636 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.)
 |-  (
 ( G  e.  W  /\  V  =  ( Base `  G )  /\  E  =  ( .ef  `  G ) )  ->  ( V UHGrph  E  <->  G  e. UHGraph  ) )
 
Theoremuhgeq12d 32637 If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph, deduction form. (Contributed by AV, 18-Jan-2020.)
 |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  ( ph  ->  ( .ef  `  G )  =  ( .ef  `  H ) )   &    |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  H  e.  Y )   =>    |-  ( ph  ->  ( G  e. UHGraph  <->  H  e. UHGraph  ) )
 
Theoremuhg0e 32638 The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  (
 ( G  e.  W  /\  ( .ef  `  G )  =  (/) )  ->  G  e. UHGraph  )
 
Theoremuhg0v 32639 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  (
 ( G  e.  W  /\  ( Base `  G )  =  (/) )  ->  ( G  e. UHGraph  <->  ( .ef  `  G )  =  (/) ) )
 
Theoremuhgrepe 32640 Replacing the edges of a hypergraph results in a hypergraph. (Contributed by AV, 18-Jan-2020.)
 |-  V  =  ( Base `  G )   &    |-  S  =  ( .ef  `  ndx )   &    |-  ( ph  ->  G  e. UHGraph  )   &    |-  ( ph  ->  E : dom  E --> ( ~P V  \  { (/) } ) )   &    |-  ( ph  ->  E  e.  _V )   =>    |-  ( ph  ->  ( G sSet  <. S ,  E >. )  e. UHGraph  )
 
Theoremuhgres 32641 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 24451. Remark: The proof is much longer (although a lot is already covered by uhgrepe 32640) than the proof of the corresponding theorem uhgrares 24435 for graphs defined as pairs. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  E  =  ( .ef  `  G )   =>    |-  ( ( G  e. UHGraph  /\  F  =  ( E  |`  A ) )  ->  ( G sSet  <. ( .ef  `  ndx ) ,  F >. )  e. UHGraph  )
 
Theoremuhgun 32642 The union of two (undirected) hypergraphs (with the same vertex set): If  <. V ,  E >. and 
<. V ,  F >. are hypergraphs, then  <. V ,  E  u.  F >. is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices), analogous to umgraun 24455. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 18-Jan-2020.)
 |-  ( ph  ->  G  e. UHGraph  )   &    |-  ( ph  ->  H  e. UHGraph  )   &    |-  E  =  ( .ef  `  G )   &    |-  F  =  ( .ef  `  H )   &    |-  S  =  ( .ef  `  ndx )   &    |-  ( ph  ->  (
 Base `  G )  =  ( Base `  H )
 )   &    |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  ( G sSet  <. S ,  ( E  u.  F ) >. )  e. UHGraph  )
 
21.25.5.5  Undirected hypergraphs (vertices)
 
Syntaxcvtx 32643 Extend class notation with the vertices of undirected hypergraphs.
 class Vtx
 
Definitiondf-vtx 32644 Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component. Therefore, this definition has no special meaning, except to give the first component of a graph its intended name. (Contributed by AV, 9-Jan-2020.)
 |- Vtx  =  ( g  e.  _V  |->  ( 1st `  g )
 )
 
Theoremvtxval 32645 The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.)
 |-  ( G  e.  V  ->  ( Vtx  `  G )  =  ( 1st `  G )
 )
 
21.25.5.6  Size and order of undirected hypergraphs
 
Syntaxcgord 32646 Extend class notation with the order of undirected hypergraphs.
 class GrOrder
 
Syntaxcgsiz 32647 Extend class notation with the size of undirected hypergraphs.
 class GrSize
 
Definitiondf-gord 32648 Define the function mapping a graph to its order. This definition is very general: It defines the order for any ordered pair as the size of its first component. In the definition of [Bollobas] p. 3, there is no special symbol for the order of a graph G, it is simply denoted by |G| and defined by |G| = |V(G)|, which corresponds to our definition, see gordopval 32652: 
( GrOrder  `  <. V ,  E >. )  =  ( # `  V ). (Contributed by AV, 3-Jan-2020.)
 |- GrOrder  =  ( g  e.  _V  |->  ( # `  ( 1st `  g
 ) ) )
 
Definitiondf-gsiz 32649 Define the function mapping a graph to its size. This definition is very general: It defines the size for any ordered pair as the size of the domain of its second component (which even needs not to be a function). In the definition of [Bollobas] p. 3, the size of a graph G is denoted by e(G) (and implicitly defined by e(G) = |E(G)|, which corresponds to our definition, see uhgraopsiz 32654:  ( GrSize  `  <. V ,  E >. )  =  ( # `  dom  E ), or usgsizedg 32657:  ( GrSize  `  G
)  =  ( # `  ( Edges  `  G ) ) ). (Contributed by AV, 3-Jan-2020.)
 |- GrSize  =  ( g  e.  _V  |->  ( # `  dom  ( 2nd `  g ) ) )
 
Theoremgordval 32650 The order of a graph. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e.  V  ->  ( GrOrder  `  G )  =  ( # `  ( 1st `  G ) ) )
 
Theoremgsizval 32651 The size of a graph. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e.  V  ->  ( GrSize  `  G )  =  ( # `  dom  ( 2nd `  G ) ) )
 
Theoremgordopval 32652 The order of a graph represented as ordered pair. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( GrOrder  `  <. V ,  E >. )  =  ( # `  V ) )
 
Theoremgsizopval 32653 The size of a graph represented as ordered pair. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( GrSize  `  <. V ,  E >. )  =  ( # `  dom  E ) )
 
Theoremuhgraopsiz 32654 The size of an undirected hypergraph as ordered pair. (Contributed by AV, 3-Jan-2020.)
 |-  ( V UHGrph  E  ->  ( GrSize  `  <. V ,  E >. )  =  ( # `  E ) )
 
Theoremuhgrasize 32655 The size of an undirected hypergraph represented as ordered pair. (Contributed by AV, 3-Jan-2020.)
 |-  ( V UHGrph  E  ->  ( V GrSize  E )  =  ( # `  E ) )
 
Theoremuhgrasiz 32656 The size of an undirected hypergraph. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e. UHGrph  ->  ( GrSize  `  G )  =  ( # `  ( 2nd `  G ) ) )
 
21.25.5.7  Edges of undirected simple graphs without loops

Theorems of subsection "Undirected simple graphs - basics" expressed with the class Edges instead of the edge function. Conventions: Since only undirected simple graphs without loops are considered in this subsection, the term "graph" is used only in this restricted meaning. The labels for the theorems will contain the acronym "usg" instead of "usgra" for short. A graph by itself is usually denoted by the class variable  G ( G  e. USGrph). The class of edges is usually denoted by the class variable  E ( E  =  ( Edges  `  G )), while a class representing an edge by itself is usually denoted by the class variable  C ( C  e.  ( Edges  `  G )) , inspired by an edge being a *c*onnection between vertices. If a a setvar variable is used for an edge, however, it is usually denoted by  e.

Acronyms used in labels: - ALT: alternative - usg: undirected simple graph without loops - edg: edge - pr: unordered pair - ppr: proper unordered pair, i e. an unordered pair containing to different sets - v: vertex - d: different - n: not - lp: loop - ad: adjacent - eu: existential uniqueness - imp: implication - inc: incident - f1: 1-1 function - lem: lemma - siz: size - ss: subset

 
Theoremusgsizedg 32657 The size of a graph is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.)
 |-  ( G  e. USGrph  ->  ( GrSize  `  G )  =  ( # `  ( Edges  `  G ) ) )
 
TheoremusgsizedgALT 32658 The size of a graph is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 12-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( G  e. USGrph  ->  ( # `  ( 2nd `  G ) )  =  ( # `
  ( Edges  `  G ) ) )
 
TheoremusgsizedgALT2 32659 The size of a graph is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  E  =  ( Edges  `  G )   &    |-  S  =  ( # `  ( 2nd `  G ) )   =>    |-  ( G  e. USGrph  ->  S  =  ( # `  E ) )
 
Theoremusgedgppr 32660 An edge of a graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge), analogous to umgrale 24448 and usgraedg2 24502. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  ( # `  C )  =  2 )
 
Theoremusgpredgv 32661 An edge of a graph always connects two vertices, analogous to usgraedgrnv 24504. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   &    |-  V  =  ( Vtx  `  G )   =>    |-  ( ( G  e. USGrph  /\ 
 { M ,  N }  e.  E )  ->  ( M  e.  V  /\  N  e.  V ) )
 
TheoremusgpredgvALT 32662 An edge of a graph always connects two vertices, analogous to usgraedgrnv 24504. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 12-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\ 
 { M ,  N }  e.  E )  ->  ( M  e.  V  /\  N  e.  V ) )
 
Theoremedgssv2ALT 32663 An edge of a graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  ( C  C_  V  /\  ( # `  C )  =  2 )
 )
 
Theoremedgssv2 32664 An edge of a graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  ( C  C_  V  /\  ( # `  C )  =  2 )
 )
 
Theoremusgedgimp 32665* If there is an edge in a graph, there are two different vertices in the graph which are connected by this edge, analogous to usgraedg3 24513 and edgprvtx 24512 and usgrarnedg 24511. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   &    |-  V  =  ( Vtx  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  E. x  e.  V  E. y  e.  V  ( x  =/=  y  /\  C  =  { x ,  y } ) )
 
Theoremusgvincvad 32666* If there is a vertex being incident with an edge in a graph, there is a(nother) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedg4 24514. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   &    |-  V  =  ( Vtx  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E. y  e.  V  C  =  { X ,  y } )
 
Theoremusgvincvadeu 32667* If there is a vertex being incident with an edge in a graph, there is exactly one (other) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedgreu 24515. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   &    |-  V  =  ( Vtx  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E! y  e.  V  C  =  { X ,  y } )
 
TheoremusgedgimpALT 32668* If there is an edge in a graph, there are two different vertices in the graph which are connected by this edge, analogous to usgraedg3 24513 and edgprvtx 24512 and usgrarnedg 24511. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  E. x  e.  V  E. y  e.  V  ( x  =/=  y  /\  C  =  { x ,  y } ) )
 
TheoremusgvincvadALT 32669* If there is a vertex being incident with an edge in a graph, there is a(nother) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedg4 24514. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E. y  e.  V  C  =  { X ,  y } )
 
TheoremusgvincvadeuALT 32670* If there is a vertex being incident with an edge in a graph, there is exactly one (other) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedgreu 24515. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E! y  e.  V  C  =  { X ,  y } )
 
Theoremusgpredgdv 32671 An edge of a graph always connects two different vertices, analogous to usgraedgrn 24508. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\ 
 { M ,  N }  e.  E )  ->  M  =/=  N )
 
Theoremusgedgnlp 32672* An edge of a graph is not a loop. (Contributed by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  -.  E. v  C  =  { v }
 )
 
Theoremusgvad2edg 32673* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex, analogous to usgra2edg 24509. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E )
 )  ->  E. x  e.  E  E. y  e.  E  ( x  =/=  y  /\  N  e.  x  /\  N  e.  y
 ) )
 
Theoremusg2edgneu 32674* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex, analogous to usgra2edg1 24510. Lemma for theorems about friendship graphs. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E )
 )  ->  -.  E! x  e.  E  N  e.  x )
 
Theoremusgedgvadf1lem1 32675* Lemma 1 for usgedgvadf1 32677. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V )
 
Theoremusgedgvadf1lem2 32676* Lemma 2 for usgedgvadf1 32677. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( M  =  (
 iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
 
Theoremusgedgvadf1 32677* The mapping of edges containing a given vertex into the set of vertices is 1-1, analogous to usgraidx2v 24520. The edge is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( e  e.  A  |->  (
 iota_ m  e.  V  e  =  { m ,  N } ) )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgedgvadf1ALTlem1 32678* Lemma 1 for usgedgvadf1 32677. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V )
 
Theoremusgedgvadf1ALTlem2 32679* Lemma 2 for usgedgvadf1 32677. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( M  =  (
 iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
 
Theoremusgedgvadf1ALT 32680* The mapping of edges containing a given vertex into the set of vertices is 1-1, analogous to usgraidx2v 24520. The edge is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( e  e.  A  |->  (
 iota_ m  e.  V  e  =  { m ,  N } ) )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgedgleord 32681* In a graph, the number of edges which contain a given vertex is not greater than the number of vertices, analogous to usgraedgleord 24521. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( # `  { e  e.  E  |  N  e.  e } )  <_  ( GrOrder  `  G ) )
 
TheoremusgedgleordALT 32682* Alternate version of usgedgleord 32681 with a shorter proof. In a graph, the number of edges which contain a given vertex is not greater than the order of the graph, i. e. the number of its vertices, analogous to usgraedgleord 24521. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 12-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  O  =  ( # `  V )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( # `  { e  e.  E  |  N  e.  e } )  <_  O )
 
21.25.5.8  Finite undirected simple graphs without loops
 
Syntaxcfusg 32683 Extend class notation with finite graphs.
 class FinUSGrph
 
Definitiondf-fusg 32684* Define the class of all finite undirected simple graphs without loops. Such a finite graph is an undirected simple graph without loops  <. V ,  E >. of finite order, i.e. where  V is finite. (Contributed by AV, 3-Jan-2020.)
 |- FinUSGrph  =  { <. v ,  e >.  |  ( e : dom  e -1-1-> { x  e.  ~P v  |  ( # `  x )  =  2 }  /\  v  e.  Fin ) }
 
Theoremrelfusgra 32685 The class of all finite undirected simple graph without loops is a relation. (Contributed by AV, 3-Jan-2020.)
 |-  Rel FinUSGrph
 
Theoremisfusgra 32686* The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V FinUSGrph  E  <->  ( E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }  /\  V  e.  Fin )
 ) )
 
Theoremisfusgra0 32687 The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V FinUSGrph  E  <->  ( V USGrph  E  /\  V  e.  Fin )
 ) )
 
Theoremisfusgracl 32688 The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e.  ( W  X.  X )  ->  ( G  e. FinUSGrph  <->  ( G  e. USGrph  /\  ( GrOrder  `  G )  e. 
 NN0 ) ) )
 
Theoremfusgraimpcl 32689 The implications of a finite undirected simple graph without loops. (Contributed by AV, 4-Jan-2020.)
 |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  ( GrOrder  `  G )  e.  NN0 ) )
 
TheoremisfusgraclALT 32690 The property of being a finite undirected simple graph without loops. (Contributed by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   =>    |-  ( G  e.  ( W  X.  X )  ->  ( G  e. FinUSGrph  <->  ( G  e. USGrph  /\  V  e.  Fin )
 ) )
 
TheoremfusgraimpclALT 32691 The implications of a finite undirected simple graph without loops. (Contributed by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   =>    |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  V  e.  Fin ) )
 
Theoremfusgusg 32692 A finite undirected simple graph without loops is a undirected simple graph without loops. (Contributed by AV, 16-Jan-2020.)
 |-  ( G  e. FinUSGrph  ->  G  e. USGrph  )
 
TheoremfusgraimpclALT2 32693 The implications of a finite undirected simple graph without loops. (Contributed by AV, 12-Jan-2020.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  ( 1st `  G )  e.  Fin ) )
 
Theoremfiusgedgfi 32694* In a finite graph the number of edges which contain a given vertex is also finite, analogous to fiusgraedgfi 24534. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  { e  e.  E  |  N  e.  e }  e.  Fin )
 
TheoremfiusgedgfiALT 32695* In a finite graph the number of edges which contain a given vertex is also finite, analogous to fiusgraedgfi 24534. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  { e  e.  E  |  N  e.  e }  e.  Fin )
 
21.25.5.9  Finite undirected simple graphs (extension)
 
Theoremusgedgffibi 32696 The number of edges in a graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.)
 |-  ( V USGrph  E  ->  ( E  e.  Fin  <->  ( V Edges  E )  e.  Fin ) )
 
Theoremusgo0s0 32697 The size of a graph of order 0 (i.e. with 0 vertices) is 0, analogous to usgrafisindb0 24535. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 4-Jan-2020.)
 |-  (
 ( G  e. USGrph  /\  ( GrOrder  `  G )  =  0 )  ->  ( GrSize  `  G )  =  0 )
 
Theoremusgo0s0ALT 32698 The size of a graph of order 0 (i.e. with 0 vertices) is 0, analogous to usgrafisindb0 24535. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  O  =  ( # `  ( 1st `  G ) )   &    |-  S  =  ( # `  ( 2nd `  G ) )   =>    |-  ( ( G  e. USGrph  /\  O  =  0 ) 
 ->  S  =  0 )
 
Theoremusgo1s0ALT 32699 The size of a graph of order 1 (i.e. with 1 vertex) is 0, analogous to usgrafisindb1 24536. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  O  =  ( # `  ( 1st `  G ) )   &    |-  S  =  ( # `  ( 2nd `  G ) )   =>    |-  ( ( G  e. USGrph  /\  O  =  1 ) 
 ->  S  =  0 )
 
Theoremusgo0fis 32700 A graph of order 0 (i.e. with 0 vertices) has a finite set of edges, analogous to usgrafisbase 24541. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.)
 |-  (
 ( G  e. USGrph  /\  ( GrOrder  `  G )  =  0 )  ->  ( Edges  `  G )  e.  Fin )
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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 38201-38213
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