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Theorem List for Metamath Proof Explorer - 39201-39300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnbgrsym 39201 A vertex in a graph is a neighbor of a second vertex iff the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.)
NeighbVtx NeighbVtx

Theoremnbumgrres 39202* The neighborhood of a vertex in a restricted multigraph (not necessarily valid for a hypergraph, because , and could be connected by one edge, so is a neighbor of in the original graph, but not in the restricted graph, because the edge between and , also incident with , was removed). (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
Vtx       Edg                     UMGraph NeighbVtx NeighbVtx

Theoremusgrnbcnvfv 39203 Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair in a simple graph. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 27-Oct-2020.)
iEdg       USGraph NeighbVtx

Theoremnbusgredgeu 39204* For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.)
Edg       USGraph NeighbVtx

Theoremedgnbusgreu 39205* For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.)
Vtx       Edg       NeighbVtx        USGraph

Theoremnbusgredgeu0 39206* For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.)
Vtx       Edg       NeighbVtx               USGraph

Theoremnbusgrf1o0 39207* The mapping of neighbors of a vertex to edges incident to the vertex is a bijection ( 1-1 onto function) in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       NeighbVtx                      USGraph

Theoremnbusgrf1o1 39208* The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       NeighbVtx               USGraph

Theoremnbusgrf1o 39209* The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       USGraph NeighbVtx

Theoremedgusgrnbfin 39210* The number of neighbors of a vertex in a simple graph is finite iff the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       USGraph NeighbVtx

Theoremnbusgrfi 39211 The class of neighbors of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       USGraph NeighbVtx

Theoremnbfiusgrfi 39212 The class of neighbors of a vertex in a finite simple graph is a finite set. (Contributed by Alexander van der Vekens, 7-Mar-2018.) (Revised by AV, 28-Oct-2020.)
FinUSGraph Vtx NeighbVtx

Theoremnb3grprlem1 39213 Lemma 1 for nb3grapr 25179. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       USGraph                      NeighbVtx

Theoremnb3grprlem2 39214* Lemma 2 for nb3grapr 25179. (Contributed by Alexander van der Vekens, 17-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       USGraph                             NeighbVtx NeighbVtx

Theoremnb3grpr 39215* The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       USGraph                             NeighbVtx

Theoremnb3grpr2 39216 The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Vtx       Edg       USGraph                             NeighbVtx NeighbVtx NeighbVtx

Theoremnb3gr2nb 39217 If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Vtx USGraph NeighbVtx NeighbVtx NeighbVtx NeighbVtx NeighbVtx

Theoremuvtxaval 39218* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
Vtx       UnivVtx NeighbVtx

Theoremuvtxael 39219* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
Vtx       UnivVtx NeighbVtx

Theoremuvtxaisvtx 39220 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
Vtx       UnivVtx

Theoremvtxnbuvtx 39221* A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
Vtx       UnivVtx NeighbVtx

Theoremuvtxanbgr 39222 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
Vtx       UnivVtx NeighbVtx

Theoremuvtxanbgrvtx 39223* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
Vtx       UnivVtx NeighbVtx

Theoremuvtxa0 39224 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
Vtx       UnivVtx

Theoremisuvtxa 39225* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
Vtx       Edg       UnivVtx

Theoremuvtxael1 39226* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Vtx       Edg       UnivVtx

Theoremuvtxa01vtx0 39227 If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by AV, 30-Oct-2020.)
Vtx       Edg       UnivVtx

Theoremuvtxa01vtx 39228 If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
Vtx       Edg       UnivVtx

Theoremuvtx2vtx1edg 39229* If a graph has two vertices, and there is an edge between the vertices, then each vertex is universal. (Contributed by AV, 1-Nov-2020.)
Vtx       Edg       UnivVtx

Theoremuvtx2vtx1edgb 39230* If a hypergraph has two vertices, there is an edge between the vertices iff each vertex is universal. (Contributed by AV, 3-Nov-2020.)
Vtx       Edg       UHGraph UnivVtx

Theoremuvtxnbgr 39231 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.)
Vtx       UnivVtx NeighbVtx

Theoremuvtxnbgrb 39232 A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)
Vtx       UnivVtx NeighbVtx

Theoremuvtxusgr 39233* The set of all universal vertices of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.)
Vtx       Edg       USGraph UnivVtx

Theoremuvtxusgrel 39234* A universal vertex, i.e. an element of the set of all universal vertices, of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.)
Vtx       Edg       USGraph UnivVtx

Theoremuvtxanm1nbgr 39235 A universal vertex has neighbors in a graph with vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.)
Vtx       FinUSGraph UnivVtx NeighbVtx

Theoremnbumgruvtxres 39236* The neighborhood of a universal vertex in a restricted multigraph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
Vtx       Edg                     UMGraph NeighbVtx NeighbVtx

Theoremuvtxumgrres 39237* A universal vertex is universal in a restricted multigraph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
Vtx       Edg                     UMGraph UnivVtx UnivVtx

Theoremiscplgr 39238* The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)
Vtx       ComplGraph UnivVtx

Theoremcplgruvtxb 39239 An graph is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 1-Nov-2020.)
Vtx       ComplGraph UnivVtx

Theoremiscplgrnb 39240* A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020.)
Vtx       ComplGraph NeighbVtx

Theoremiscplgredg 39241* A graph is complete iff all vertices are connected with each other by (at least) one edge. (Contributed by AV, 10-Nov-2020.)
Vtx       Edg       ComplGraph

Theoremiscusgr 39242 The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
ComplUSGraph USGraph ComplGraph

Theoremcusgrusgr 39243 A complete simple graph is a simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
ComplUSGraph USGraph

Theoremcusgrcplgr 39244 A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.)
ComplUSGraph ComplGraph

Theoremiscusgrvtx 39245* A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.)
Vtx       ComplUSGraph USGraph UnivVtx

Theoremcusgruvtxb 39246 A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.)
Vtx       USGraph ComplUSGraph UnivVtx

Theoremiscusgredg 39247* A simple graph is complete iff all vertices are connected by an edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 1-Nov-2020.)
Vtx       Edg       ComplUSGraph USGraph

Theoremcusgredg 39248* In a complete simple graph, the edges are all the pairs of different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 1-Nov-2020.)
Vtx       Edg       ComplUSGraph

Theoremcplgr0 39249 The null graph (with no vertices and no edges) represented by the empty set is a complete graph. (Contributed by AV, 1-Nov-2020.)
ComplGraph

Theoremcusgr0 39250 The null graph (with no vertices and no edges) represented by the empty set is a complete simple graph. (Contributed by AV, 1-Nov-2020.)
ComplUSGraph

Theoremcplgr0v 39251 A graph with no vertices (and therefore no edges) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
Vtx       ComplGraph

Theoremcusgr0v 39252 A graph with no vertices (and therefore no edges) is a complete simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
Vtx       iEdg ComplUSGraph

Theoremcplgr1v 39253 A graph with one vertex is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
Vtx       ComplGraph

Theoremcusgr1v 39254 A graph with one vertex and no edges is a complete simple graph. (Contributed by AV, 1-Nov-2020.)
Vtx       iEdg ComplUSGraph

Theoremcplgr2v 39255 An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020.)
Vtx       Edg       UHGraph ComplGraph

Theoremcplgr2vpr 39256 An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.) (Revised by AV, 3-Nov-2020.)
Vtx       Edg       UHGraph ComplGraph

Theoremnbcplgr 39257 In a complete graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)
Vtx       ComplGraph NeighbVtx

Theoremcplgr3v 39258 A multigraph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.)
Edg       Vtx        UMGraph ComplGraph

Theoremcusgr3vnbpr 39259* The neighbors of a vertex in a simple graph with three elements are unordered pairs of the other vertices if and only if the graph is complete. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 5-Nov-2020.)
Edg       Vtx        Vtx       USGraph ComplGraph NeighbVtx

Theoremcplgrop 39260 A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
ComplGraph Vtx iEdg ComplGraph

Theoremcusgrop 39261 A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
ComplUSGraph Vtx iEdg ComplUSGraph

Theoremusgrexi 39262* An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
USGraph

Theoremcusgrexi 39263* An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
ComplUSGraph

Theoremcusgrexg 39264* For each set there is a set of edges so that the set together with these edges is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
ComplUSGraph

Theoremcusgrres 39265* Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
Vtx       Edg                     ComplUSGraph ComplUSGraph

Theoremcusgrsizeindb0 39266 Base case of the induction in cusgrasize 25204. The size of a complete simple graph with 0 vertices, actually of every null graph, is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.)
Vtx       Edg       UHGraph

Theoremcusgrsizeindb1 39267 Base case of the induction in cusgrasize 25204. The size of a (complete) simple graph with 1 vertex is 0=((1-1)*1)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.)
Vtx       Edg       USGraph

Theoremcusgrsizeindslem 39268* Lemma for cusgrsizeinds 39269. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
Vtx       Edg       ComplUSGraph

Theoremcusgrsizeinds 39269* Part 1 of induction step in cusgrsize 39271. The size of a complete simple graph with vertices is plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
Vtx       Edg              ComplUSGraph

Theoremcusgrsize2inds 39270* Induction step in cusgrasize 25204. If the size of the complete graph with vertices reduced by one vertex is " choose 2", the size of the complete graph with vertices is " choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
Vtx       Edg              ComplUSGraph

Theoremcusgrsize 39271 The size of a finite complete simple graph with vertices ( ) is (" choose 2") resp. , see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)
Vtx       Edg       ComplUSGraph

Theoremcusgrfilem1 39272* Lemma 1 for cusgrfi 39275. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
Vtx              ComplUSGraph Edg

Theoremcusgrfilem2 39273* Lemma 2 for cusgrfi 39275. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
Vtx

Theoremcusgrfilem3 39274* Lemma 3 for cusgrfi 39275. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
Vtx

Theoremcusgrfi 39275 If the size of a complete simple graph is finite, then its order is also finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
Vtx       Edg       ComplUSGraph

Theoremusgredgsscusgredg 39276 A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
Vtx       Edg       Vtx       Edg       USGraph ComplUSGraph

Theoremusgrsscusgr 39277* A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
Vtx       Edg       Vtx       Edg       USGraph ComplUSGraph

Theoremsizusglecusglem1 39278 Lemma 1 for sizusglecusg 39280. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 13-Nov-2020.)
Vtx       Edg       Vtx       Edg       USGraph ComplUSGraph

Theoremsizusglecusglem2 39279 Lemma 2 for sizusglecusg 39280. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
Vtx       Edg       Vtx       Edg       USGraph ComplUSGraph

Theoremsizusglecusg 39280 The size of a simple graph with vertices is at most the size of a complete simple graph with vertices ( may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
Vtx       Edg       Vtx       Edg       USGraph ComplUSGraph

Theoremfusgrmaxsize 39281 The maximum size of a finite simple graph with vertices is . See statement in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 14-Nov-2020.)
Vtx       Edg       FinUSGraph

21.33.9  Graph theory (old)

21.33.9.1  Undirected hypergraphs

Theoremuhgraedgrnv 39282 An edge of an undirected hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
UHGrph

21.33.9.2  Walks, Paths and Cycles

Theoremwlkc 39283* A walk as class. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
Walks Walks

Theoremusgra2pthspth 39284 In a undirected simple graph, any path of length 2 is a simple path. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
USGrph Paths SPaths

Theoremspthdifv 39285 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.)
SPaths

Theoremusgra2pthlem1 39286* Lemma for usgra2pth 39287. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
..^ ..^ USGrph

Theoremusgra2pth 39287* 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.)
USGrph Paths ..^

Theoremusgra2pth0 39288* 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.)
USGrph Paths ..^

Theoremusgra2adedglem1 39289 In an undirected simple graph, two adjacent edges are an unordered pair of unordered pairs. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
USGrph

21.33.9.3  Vertex degree (extension)

Theoremvdusgravaledg 39290* The value of the vertex degree function for simple undirected graphs in terms of edges. (Contributed by Alexander van der Vekens, 9-Jul-2018.)
USGrph VDeg

Theoremusgrauvtxvd 39291 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.)
USGrph UnivVertex VDeg

Theoremvdcusgra 39292* 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.)
ComplUSGrph VDeg

21.33.9.4  Undirected hypergraphs as extensible structures

Syntaxcuhgraltv 39293 Extend class notation with undirected hypergraphs as extensible structures.
UHGraphALTV

Syntaxcushgraltv 39294 Extend class notation with undirected simple hypergraphs as extensible structures.
USHGraphALTV

Definitiondf-uhgrALTV 39295* 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.)
UHGraphALTV .ef

Definitiondf-ushgrALTV 39296* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph where is an injective (one-to-one) function into subsets of , 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.)
USHGraphALTV .ef

TheoremisuhgrALTV 39297 The predicate "is an undirected hypergraph." (Contributed by AV, 18-Jan-2020.)
.ef       UHGraphALTV

TheoremisushgrALTV 39298 The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.)
.ef       USHGraphALTV

TheoremuhgfALTV 39299 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.)
.ef       UHGraphALTV

TheoremuhgssALTV 39300 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
.ef       UHGraphALTV

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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