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Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremusgr0op 39101 The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
 |-  ( V  e.  W  ->  <. V ,  (/) >.  e. USGraph  )
 
Theoremuspgr1op 39102 The simple pseudograph with one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  <. V ,  { <. A ,  { B ,  C } >. } >.  e. USPGraph  )
 
Theoremusgr1op 39103 The simple graph with one edge (with additional assumption that  B  =/=  C since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  ( B  =/=  C  ->  <. V ,  { <. A ,  { B ,  C } >. } >.  e. USGraph  ) )
 
Theoremedg0usgr 39104 A class without edges is a simple graph. Since  ran 
F  =  (/) does not generally imply  Fun  F, but  Fun  (iEdg `  G ) is required for  G to be a graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.)
 |-  (
 ( G  e.  W  /\  (Edg `  G )  =  (/)  /\  Fun  (iEdg `  G ) )  ->  G  e. USGraph  )
 
Theoremusgr1vr 39105 A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.)
 |-  (
 ( G  e.  W  /\  A  e.  X  /\  (Vtx `  G )  =  { A } )  ->  ( G  e. USGraph  ->  (iEdg `  G )  =  (/) ) )
 
Theoremusgr1v 39106 A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  { A } )  ->  ( G  e. USGraph  <->  (iEdg `  G )  =  (/) ) )
 
Theoremusgr1v0edg 39107 A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  { A }  /\  Fun  (iEdg `  G )
 )  ->  ( G  e. USGraph  <-> 
 (Edg `  G )  =  (/) ) )
 
Theoremusgrexmpllem 39108 Lemma for usgrexmpl 39111. (Contributed by AV, 21-Oct-2020.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  ( (Vtx `  G )  =  V  /\  (iEdg `  G )  =  E )
 
Theoremusgrexmplvtx 39109 The vertices  0 ,  1 ,  2 ,  3 ,  4 of the graph  G  =  <. V ,  E >.. (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  (Vtx `  G )  =  ( { 0 ,  1 ,  2 }  u.  { 3 ,  4 } )
 
Theoremusgrexmpledg 39110 The edges  { 0 ,  1 } ,  {
1 ,  2 } ,  { 2 ,  0 } ,  {
0 ,  3 } of the graph  G  =  <. V ,  E >.. (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  (Edg `  G )  =  ( { { 0 ,  1 } ,  { 1 ,  2 } }  u.  { { 2 ,  0 } ,  { 0 ,  3 } }
 )
 
Theoremusgrexmpl 39111  G is a simple graph of five vertices  0 ,  1 ,  2 ,  3 ,  4, with edges  { 0 ,  1 } ,  {
1 ,  2 } ,  { 2 ,  0 } ,  {
0 ,  3 }. (Contributed by Alexander van der Vekens, 15-Aug-2017.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   &    |-  G  =  <. V ,  E >.   =>    |-  G  e. USGraph
 
21.33.8.7  Subgraphs
 
Syntaxcsubgr 39112 Extend class notation with subgraphs.
 class SubGraph
 
Definitiondf-subgr 39113* Define the class of the subgraph relation. A class  s is a subgraph of a class  g (the supergraph of 
s) if its vertices are also vertices of  g, and its edges are also edges of  g, connecting vertices of  s only (see section I.1 in [Bollobas] p. 2 or section 1.1 in [Diestel] p. 4). The second condition is ensured by the requirement that the edge function of  s is a restriction of the edge function of  g having only vertices of  s in its range. Note that the domains of the edge functions of the subgraph and the supergraph should be compatible. (Contributed by AV, 16-Nov-2020.)
 |- SubGraph  =  { <. s ,  g >.  |  ( (Vtx `  s
 )  C_  (Vtx `  g
 )  /\  (iEdg `  s
 )  =  ( (iEdg `  g )  |`  dom  (iEdg `  s ) )  /\  (Edg `  s )  C_  ~P (Vtx `  s )
 ) }
 
Theoremrelsubgr 39114 The class of the subgraph relation is a relation. (Contributed by AV, 16-Nov-2020.)
 |-  Rel SubGraph
 
Theoremsubgrv 39115 If a class is a subgraph of another class, both classes are sets. (Contributed by AV, 16-Nov-2020.)
 |-  ( S SubGraph  G  ->  ( S  e.  _V  /\  G  e.  _V ) )
 
Theoremissubgr 39116 The property of a set to be a subgraph of another set. (Contributed by AV, 16-Nov-2020.)
 |-  V  =  (Vtx `  S )   &    |-  A  =  (Vtx `  G )   &    |-  I  =  (iEdg `  S )   &    |-  B  =  (iEdg `  G )   &    |-  E  =  (Edg `  S )   =>    |-  (
 ( G  e.  W  /\  S  e.  U ) 
 ->  ( S SubGraph  G  <->  ( V  C_  A  /\  I  =  ( B  |`  dom  I ) 
 /\  E  C_  ~P V ) ) )
 
Theoremissubgr2 39117 The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-Nov-2020.)
 |-  V  =  (Vtx `  S )   &    |-  A  =  (Vtx `  G )   &    |-  I  =  (iEdg `  S )   &    |-  B  =  (iEdg `  G )   &    |-  E  =  (Edg `  S )   =>    |-  (
 ( G  e.  W  /\  Fun  B  /\  S  e.  U )  ->  ( S SubGraph  G  <->  ( V  C_  A  /\  I  C_  B  /\  E  C_  ~P V ) ) )
 
Theoremsubgrprop 39118 The properties of a subgraph. (Contributed by AV, 19-Nov-2020.)
 |-  V  =  (Vtx `  S )   &    |-  A  =  (Vtx `  G )   &    |-  I  =  (iEdg `  S )   &    |-  B  =  (iEdg `  G )   &    |-  E  =  (Edg `  S )   =>    |-  ( S SubGraph  G  ->  ( V  C_  A  /\  I  =  ( B  |`  dom  I
 )  /\  E  C_  ~P V ) )
 
Theoremsubgrprop2 39119 The properties of a subgraph: If 
S is a subgraph of  G, its vertices are also vertices of  G, and its edges are also edges of  G, connecting vertices of the subgraph only. (Contributed by AV, 19-Nov-2020.)
 |-  V  =  (Vtx `  S )   &    |-  A  =  (Vtx `  G )   &    |-  I  =  (iEdg `  S )   &    |-  B  =  (iEdg `  G )   &    |-  E  =  (Edg `  S )   =>    |-  ( S SubGraph  G  ->  ( V  C_  A  /\  I  C_  B  /\  E  C_  ~P V ) )
 
Theoremuhgrissubgr 39120 The property of a hypergraph to be a subgraph. (Contributed by AV, 19-Nov-2020.)
 |-  V  =  (Vtx `  S )   &    |-  A  =  (Vtx `  G )   &    |-  I  =  (iEdg `  S )   &    |-  B  =  (iEdg `  G )   =>    |-  (
 ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  ->  ( S SubGraph  G  <-> 
 ( V  C_  A  /\  I  C_  B ) ) )
 
Theoremsubgrprop3 39121 The properties of a subgraph: If 
S is a subgraph of  G, its vertices are also vertices of  G, and its edges are also edges of  G. (Contributed by AV, 19-Nov-2020.)
 |-  V  =  (Vtx `  S )   &    |-  A  =  (Vtx `  G )   &    |-  E  =  (Edg `  S )   &    |-  B  =  (Edg `  G )   =>    |-  ( S SubGraph  G  ->  ( V  C_  A  /\  E  C_  B ) )
 
Theoremegrsubgr 39122 An empty graph consisting of a subset of vertices of a graph (and having no edges) is a subgraph of the graph. (Contributed by AV, 17-Nov-2020.)
 |-  (
 ( ( G  e.  W  /\  S  e.  U )  /\  (Vtx `  S )  C_  (Vtx `  G )  /\  ( Fun  (iEdg `  S )  /\  (Edg `  S )  =  (/) ) )  ->  S SubGraph  G )
 
Theorem0grsubgr 39123 The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
 |-  ( G  e.  W  ->  (/) SubGraph  G )
 
Theorem0uhgrsubgr 39124 The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
 |-  (
 ( G  e.  W  /\  S  e. UHGraph  /\  (Vtx `  S )  =  (/) )  ->  S SubGraph  G )
 
Theoremuhgrsubgrself 39125 A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
 |-  ( G  e. UHGraph  ->  G SubGraph  G )
 
Theoremsubgrfun 39126 The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020.)
 |-  (
 ( Fun  (iEdg `  G )  /\  S SubGraph  G )  ->  Fun  (iEdg `  S ) )
 
Theoremsubgruhgrfun 39127 The edge function of a subgraph of a hypergraph is a function. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 20-Nov-2020.)
 |-  (
 ( G  e. UHGraph  /\  S SubGraph  G )  ->  Fun  (iEdg `  S ) )
 
Theoremsubgreldmiedg 39128 An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.)
 |-  (
 ( S SubGraph  G  /\  X  e.  dom  (iEdg `  S ) )  ->  X  e.  dom  (iEdg `  G )
 )
 
Theoremsubgruhgredgd 39129 An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)
 |-  V  =  (Vtx `  S )   &    |-  I  =  (iEdg `  S )   &    |-  ( ph  ->  G  e. UHGraph  )   &    |-  ( ph  ->  S SubGraph  G )   &    |-  ( ph  ->  X  e.  dom  I )   =>    |-  ( ph  ->  ( I `  X )  e.  ( ~P V  \  { (/) } ) )
 
Theoremsubuhgr 39130 A subgraph of a hypergraph is a hypergraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
 |-  (
 ( G  e. UHGraph  /\  S SubGraph  G )  ->  S  e. UHGraph  )
 
Theoremsubumgr 39131 A subgraph of a multigraph is a multigraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
 |-  (
 ( G  e. UMGraph  /\  S SubGraph  G )  ->  S  e. UMGraph  )
 
Theoremsubusgr 39132 A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
 |-  (
 ( G  e. USGraph  /\  S SubGraph  G )  ->  S  e. USGraph  )
 
Theoremuhgrspansubgrlem 39133 Lemma for uhgrspansubgr 39134: The edges of the graph  S obtained by removing some edges of a hypergraph  G are subsets of its vertices (a spanning subgraph, see comment for uhgrspansubgr 39134. (Contributed by AV, 18-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  (Vtx `  S )  =  V )   &    |-  ( ph  ->  (iEdg `  S )  =  ( E  |`  A )
 )   &    |-  ( ph  ->  G  e. UHGraph  )   =>    |-  ( ph  ->  (Edg `  S )  C_  ~P (Vtx `  S ) )
 
Theoremuhgrspansubgr 39134 A spanning subgraph  S of a hypergraph  G is actually a subgraph of  G. A subgraph  S of a graph  G which has the same vertices as  G and is obtained by removing some edges of  G is called a spanning subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). Formally, the edges are "removed" by restricting the edge function of the original graph by an arbitrary class (which actually needs not to be a subset of the domain of the edge function). (Contributed by AV, 18-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  (Vtx `  S )  =  V )   &    |-  ( ph  ->  (iEdg `  S )  =  ( E  |`  A )
 )   &    |-  ( ph  ->  G  e. UHGraph  )   =>    |-  ( ph  ->  S SubGraph  G )
 
Theoremuhgrspan 39135 A spanning subgraph  S of a hypergraph  G is a hypergraph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  (Vtx `  S )  =  V )   &    |-  ( ph  ->  (iEdg `  S )  =  ( E  |`  A )
 )   &    |-  ( ph  ->  G  e. UHGraph  )   =>    |-  ( ph  ->  S  e. UHGraph  )
 
Theoremumgrspan 39136 A spanning subgraph  S of a mulitgraph  G is a multigraph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  (Vtx `  S )  =  V )   &    |-  ( ph  ->  (iEdg `  S )  =  ( E  |`  A )
 )   &    |-  ( ph  ->  G  e. UMGraph  )   =>    |-  ( ph  ->  S  e. UMGraph  )
 
Theoremusgrspan 39137 A spanning subgraph  S of a simple graph  G is a simple graph. (Contributed by AV, 15-Oct-2020.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  (Vtx `  S )  =  V )   &    |-  ( ph  ->  (iEdg `  S )  =  ( E  |`  A )
 )   &    |-  ( ph  ->  G  e. USGraph  )   =>    |-  ( ph  ->  S  e. USGraph  )
 
Theoremuhgrspanop 39138 A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. UHGraph  ->  <. V ,  ( E  |`  A )
 >.  e. UHGraph  )
 
Theoremumgrspanop 39139 A spanning subgraph of a multigraph represented by an ordered pair is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. UMGraph  ->  <. V ,  ( E  |`  A )
 >.  e. UMGraph  )
 
Theoremusgrspanop 39140 A spanning subgraph of a simple graph represented by an ordered pair is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (iEdg `  G )   =>    |-  ( G  e. USGraph  ->  <. V ,  ( E  |`  A )
 >.  e. USGraph  )
 
Theoremuhgrspan1lem1 39141 Lemma 1 for uhgrspan1 39144. (Contributed by AV, 19-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  { i  e.  dom  I  |  N  e/  ( I `  i ) }   =>    |-  (
 ( V  \  { N } )  e.  _V  /\  ( I  |`  F )  e.  _V )
 
Theoremuhgrspan1lem2 39142 Lemma 2 for uhgrspan1 39144. (Contributed by AV, 19-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  { i  e.  dom  I  |  N  e/  ( I `  i ) }   &    |-  S  =  <. ( V  \  { N } ) ,  ( I  |`  F )
 >.   =>    |-  (Vtx `  S )  =  ( V  \  { N } )
 
Theoremuhgrspan1lem3 39143 Lemma 3 for uhgrspan1 39144. (Contributed by AV, 19-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  { i  e.  dom  I  |  N  e/  ( I `  i ) }   &    |-  S  =  <. ( V  \  { N } ) ,  ( I  |`  F )
 >.   =>    |-  (iEdg `  S )  =  ( I  |`  F )
 
Theoremuhgrspan1 39144* The induced subgraph  S of a hypergraph  G obtained by removing one vertex is actually a subgraph of  G. A subgraph is called induced or spanned by a subset of vertices of a graph if it contains all edges of the original graphs that join two vertices of the subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). (Contributed by AV, 19-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  { i  e.  dom  I  |  N  e/  ( I `  i ) }   &    |-  S  =  <. ( V  \  { N } ) ,  ( I  |`  F )
 >.   =>    |-  ( ( G  e. UHGraph  /\  N  e.  V ) 
 ->  S SubGraph  G )
 
Theoremumgrres1lem1 39145* Lemma 1 for umgrres1 39148. (Contributed by AV, 7-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  (
 ( V  \  { N } )  e.  _V  /\  (  _I  |`  F )  e.  _V )
 
Theoremumgrres1lem2 39146* Lemma 2 for umgrres1 39148. (Contributed by AV, 7-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   &    |-  S  =  <. ( V  \  { N } ) ,  (  _I  |`  F )
 >.   =>    |-  (Vtx `  S )  =  ( V  \  { N } )
 
Theoremumgrres1lem3 39147* Lemma 3 for umgrres1 39148. (Contributed by AV, 7-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   &    |-  S  =  <. ( V  \  { N } ) ,  (  _I  |`  F )
 >.   =>    |-  (iEdg `  S )  =  (  _I  |`  F )
 
Theoremumgrres1 39148* A graph obtained by removing one vertex and all edges incident with this vertex is a multigraph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 39113 since the domains of the edge functions may not be compatible. (Contributed by AV, 8-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   &    |-  S  =  <. ( V  \  { N } ) ,  (  _I  |`  F )
 >.   =>    |-  ( ( G  e. UMGraph  /\  N  e.  V ) 
 ->  S  e. UMGraph  )
 
Theoremusgrres1 39149* Restricting a simple graph by removing one vertex results in a simple graph. Remark: This restricted graph is not a subgraph of the original graph in the sense of df-subgr 39113 since the domains of the edge functions may not be compatible. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 8-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   &    |-  S  =  <. ( V  \  { N } ) ,  (  _I  |`  F )
 >.   =>    |-  ( ( G  e. USGraph  /\  N  e.  V ) 
 ->  S  e. USGraph  )
 
21.33.8.8  Undirected simple graphs - finite graphs
 
Syntaxcfusgr 39150 Extend class notation with finite simple graphs.
 class FinUSGraph
 
Definitiondf-fusgr 39151 Define the class of all finite undirected simple graphs without loops (called "finite simple graphs" in the following). A finite graph is an undirected simple graph of finite order, i.e. with a finite set of vertices. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
 |- FinUSGraph  =  {
 g  e. USGraph  |  (Vtx `  g )  e.  Fin }
 
Theoremisfusgr 39152 The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. FinUSGraph  <->  ( G  e. USGraph  /\  V  e.  Fin )
 )
 
Theoremisfusgrf1 39153* The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( G  e.  W  ->  ( G  e. FinUSGraph  <->  ( I : dom  I -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }  /\  V  e.  Fin )
 ) )
 
Theoremfusgrusgr 39154 A finite simple graph is a simple graph. (Contributed by AV, 16-Jan-2020.) (Revised by AV, 21-Oct-2020.)
 |-  ( G  e. FinUSGraph  ->  G  e. USGraph  )
 
Theoremopfusgr 39155 A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( <. V ,  E >.  e. FinUSGraph 
 <->  ( <. V ,  E >.  e. USGraph  /\  V  e.  Fin ) ) )
 
Theoremusgredgffibi 39156 The number of edges in a simple graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 22-Oct-2020.)
 |-  I  =  (iEdg `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. USGraph  ->  ( E  e.  Fin  <->  I  e.  Fin ) )
 
Theoremfusgredgfi 39157* In a finite simple graph the number of edges which contain a given vertex is also finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 21-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. FinUSGraph  /\  N  e.  V )  ->  { e  e.  E  |  N  e.  e }  e.  Fin )
 
Theoremusgr1v0e 39158 The size of a (finite) simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 22-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  ( # `
  V )  =  1 )  ->  ( # `
  E )  =  0 )
 
Theoremusgrfilem 39159* In a finite simple graph, the number of edges is finite iff the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  (
 ( G  e. FinUSGraph  /\  N  e.  V )  ->  ( E  e.  Fin  <->  F  e.  Fin ) )
 
Theoremfusgrfisbase 39160 Induction base for fusgrfis 39162. Main work is done in uhgr0v0e 39095. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  <. V ,  E >.  e. USGraph  /\  ( # `  V )  =  0 )  ->  E  e.  Fin )
 
Theoremfusgrfisstep 39161* Induction step in fusgrfis 39162: In a finite simple graph, the number of edges is finite if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  <. V ,  E >.  e. FinUSGraph  /\  N  e.  V )  ->  ( (  _I  |`  { p  e.  (Edg ` 
 <. V ,  E >. )  |  N  e/  p } )  e.  Fin  ->  E  e.  Fin ) )
 
Theoremfusgrfis 39162 A finite simple graph is of finite size, i.e. has a finite number of edges. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 8-Nov-2020.)
 |-  ( G  e. FinUSGraph  ->  (Edg `  G )  e.  Fin )
 
21.33.8.9  Neighbors, complete graphs and universal vertices
 
Syntaxcnbgr 39163 Extend class notation with neighbors (of a vertex in a graph).
 class NeighbVtx
 
Syntaxcuvtxa 39164 Extend class notation with the universal vertices (in a graph).
 class UnivVtx
 
Syntaxccplgr 39165 Extend class notation with (arbitrary) complete graphs.
 class ComplGraph
 
Syntaxccusgr 39166 Extend class notation with complete simple graphs.
 class ComplUSGraph
 
Definitiondf-nbgr 39167* Define the (open) neighborhood resp. the class of all neighbors of a vertex (in a graph), see definition in section I.1 of [Bollobas] p. 3 or definition in section 1.1 of [Diestel] p. 3. The neighborhood/neighbors of a vertex are all (other) vertices which are connected with this vertex by an edge. In contrast to a closed neighborhood, a vertex is not a neighbor of itself. This definition is applicable even for arbitrary hypergraphs.

Remark: To distinguish this definition from other definitions for neighborhoods resp. neighbors (e.g.  nei in Topology, see df-nei 20112), the suffix Vtx is added to the class constant NeighbVtx. (Contributed by Alexander van der Vekens and Mario Carneiro, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.)

 |- NeighbVtx  =  ( g  e.  _V ,  v  e.  (Vtx `  g
 )  |->  { n  e.  (
 (Vtx `  g )  \  { v } )  |  E. e  e.  (Edg `  g ) { v ,  n }  C_  e } )
 
Theoremnbgrprc0 39168 The set of neighbors is empty if the graph or the vertex are proper classes. (Contributed by AV, 26-Oct-2020.)
 |-  ( -.  ( G  e.  _V  /\  N  e.  _V )  ->  ( G NeighbVtx  N )  =  (/) )
 
Definitiondf-uvtxa 39169* Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph) resp. all other vertices are its neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.)
 |- UnivVtx  =  ( g  e.  _V  |->  { v  e.  (Vtx `  g )  |  A. n  e.  ( (Vtx `  g
 )  \  { v } ) n  e.  ( g NeighbVtx  v ) } )
 
Definitiondf-cplgr 39170* Define the class of all complete "graphs". A class/graph is called complete if every pair of distinct vertices is connected by an edge, i.e. each vertex has all other vertices as neighbors. (Contributed by AV, 24-Oct-2020.)
 |- ComplGraph  =  {
 g  |  A. v  e.  (Vtx `  g )
 v  e.  (UnivVtx `  g
 ) }
 
Definitiondf-cusgr 39171 Define the class of all complete simple graphs. A simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge, see definition in section 1.1 of [Diestel] p. 3. In contrast, the definition in section I.1 of [Bollobas] p. 3 is based on the size of (finite) complete graphs, see cusgrsize 39271. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.)
 |- ComplUSGraph  =  {
 g  e. USGraph  |  g  e. ComplGraph }
 
Theoremnbgrval 39172* The set of neighbors of a vertex  V in a graph  G. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e.  W  /\  N  e.  V ) 
 ->  ( G NeighbVtx  N )  =  { n  e.  ( V  \  { N }
 )  |  E. e  e.  E  { N ,  n }  C_  e }
 )
 
Theoremdfnbgr2 39173* Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e.  W  /\  N  e.  V ) 
 ->  ( G NeighbVtx  N )  =  { n  e.  ( V  \  { N }
 )  |  E. e  e.  E  ( N  e.  e  /\  n  e.  e
 ) } )
 
Theoremdfnbgr3 39174* Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves [see also nbgraop1 25151]. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e.  W  /\  N  e.  V  /\  Fun 
 I )  ->  ( G NeighbVtx  N )  =  { n  e.  ( V  \  { N } )  |  E. i  e.  dom  I { N ,  n }  C_  ( I `  i ) } )
 
Theoremnbgrnvtx0 39175 There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( N  e/  V  ->  ( G NeighbVtx  N )  =  (/) )
 
Theoremnbgrel 39176* Characterization of a neighbor of a vertex  V in a graph  G. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e.  W  ->  ( K  e.  ( G NeighbVtx  N )  <->  ( ( K  e.  V  /\  N  e.  V )  /\  K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) )
 
Theoremnbuhgr 39177* The set of neighbors of a vertex in a hypergraph. This version of nbgrval 39172 (with  N being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  N  e.  X )  ->  ( G NeighbVtx  N )  =  { n  e.  ( V  \  { N } )  |  E. e  e.  E  { N ,  n }  C_  e } )
 
Theoremnbumgr 39178* The set of neighbors of a vertex in a multigraph. (Contributed by AV, 5-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  N  e.  V )  ->  ( G NeighbVtx  N )  =  { n  e.  ( V  \  { N } )  |  { N ,  n }  e.  E }
 )
 
Theoremnbumgrel 39179 A neighbor of a vertex in a multigraph. (Contributed by AV, 5-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( ( G  e. UMGraph  /\  K  e.  V ) 
 /\  ( N  e.  V  /\  N  =/=  K ) )  ->  ( N  e.  ( G NeighbVtx  K )  <->  { N ,  K }  e.  E ) )
 
Theoremnbusgrvtx 39180* The set of neighbors of a vertex in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  N  e.  V )  ->  ( G NeighbVtx  N )  =  { n  e.  V  |  { N ,  n }  e.  E } )
 
Theoremnbusgr 39181* The set of neighbors of an arbitrary class in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. USGraph  ->  ( G NeighbVtx  N )  =  { n  e.  V  |  { N ,  n }  e.  E } )
 
Theoremnbgr2vtx1edg 39182* If a graph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e.  W  /\  ( # `  V )  =  2  /\  V  e.  E )  ->  A. v  e.  V  A. n  e.  ( V 
 \  { v }
 ) n  e.  ( G NeighbVtx  v ) )
 
Theoremnbuhgr2vtx1edgblem 39183* Lemma for nbuhgr2vtx1edgb 39184. This reverse direction of nbgr2vtx1edg 39182 only holds for classes whose edges are subsets of the set of vertices (hypergraphs!) (Contributed by AV, 2-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  V  =  { a ,  b }  /\  a  e.  ( G NeighbVtx  b ) )  ->  { a ,  b }  e.  E )
 
Theoremnbuhgr2vtx1edgb 39184* If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  ( # `
  V )  =  2 )  ->  ( V  e.  E  <->  A. v  e.  V  A. n  e.  ( V 
 \  { v }
 ) n  e.  ( G NeighbVtx  v ) ) )
 
Theoremnbusgreledg 39185 A class/vertex is a neighbor of another class/vertex in a simple graph iff the vertices are endpoints of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.) (Revised by AV, 26-Oct-2020.)
 |-  E  =  (Edg `  G )   =>    |-  ( G  e. USGraph  ->  ( N  e.  ( G NeighbVtx  K )  <->  { N ,  K }  e.  E ) )
 
Theoremuhgrnbgr0nb 39186* A vertex which is not endpoint of an edge has no neighbor in a hypergraph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
 |-  (
 ( G  e. UHGraph  /\  A. e  e.  (Edg `  G ) N  e/  e ) 
 ->  ( G NeighbVtx  N )  =  (/) )
 
Theoremnbgr0vtxlem 39187* Lemma for nbgr0vtx 39188 and nbgr0edg 39189. (Contributed by AV, 15-Nov-2020.)
 |-  ( ph  ->  A. n  e.  (
 (Vtx `  G )  \  { K } )  -.  E. e  e.  (Edg `  G ) { K ,  n }  C_  e
 )   =>    |-  ( ph  ->  ( G NeighbVtx  K )  =  (/) )
 
Theoremnbgr0vtx 39188 In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
 |-  (
 (Vtx `  G )  =  (/)  ->  ( G NeighbVtx  K )  =  (/) )
 
Theoremnbgr0edg 39189 In an empty graph (with no edges), every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
 |-  (
 (Edg `  G )  =  (/)  ->  ( G NeighbVtx  K )  =  (/) )
 
Theoremnbgr1vtx 39190 In a graph with one vertex, all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
 |-  (
 ( # `  (Vtx `  G ) )  =  1  ->  ( G NeighbVtx  K )  =  (/) )
 
Theoremnbgrisvtx 39191 Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  N  e.  ( G NeighbVtx  K ) )  ->  N  e.  V )
 
Theoremnbgrssvtx 39192 The neighbors of a vertex in a graph are a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  ( G NeighbVtx  N )  C_  V )
 
Theoremnbgrnself 39193* A vertex in a graph is not a neighbor of itself. (Contributed by by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  A. v  e.  V  v 
 e/  ( G NeighbVtx  v ) )
 
Theoremusgrnbnself 39194* A vertex in a simple graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. USGraph  ->  A. v  e.  V  v  e/  ( G NeighbVtx  v ) )
 
Theoremnbgrnself2 39195 A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)
 |-  ( G  e.  W  ->  N 
 e/  ( G NeighbVtx  N ) )
 
Theoremnbgrssovtx 39196 The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 39192. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e.  W  ->  ( G NeighbVtx  N )  C_  ( V  \  { N }
 ) )
 
Theoremnbgrssvwo2 39197 The neighbors of a vertex are a subset of all vertices except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  M  e/  ( G NeighbVtx  N ) )  ->  ( G NeighbVtx  N )  C_  ( V  \  { M ,  N } ) )
 
Theoremusgrnbnself2 39198 In a simple graph, a class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.)
 |-  ( G  e. USGraph  ->  N  e/  ( G NeighbVtx  N ) )
 
Theoremusgrnbssovtx 39199 The neighbors of a vertex in a simple graph are a subset of all vertices of the graph except the vertex itself. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. USGraph  ->  ( G NeighbVtx  N )  C_  ( V 
 \  { N }
 ) )
 
Theoremusgrnbssvwo2 39200 The neighbors of a vertex in a simple graph are a subset of all vertices of the graph except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 27-Oct-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. USGraph  /\  M  e/  ( G NeighbVtx  N )
 )  ->  ( G NeighbVtx  N )  C_  ( V  \  { M ,  N } ) )
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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 40101-40161
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