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Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisusgr 39401* The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USGraph

Theoremuspgrf 39402* The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USPGraph

Theoremusgrf 39403* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USGraph

Theoremisusgrs 39404* The property of being a simple graph, simplified version of isusgr 39401. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.) (Proof shortened by AV, 24-Nov-2020.)
Vtx       iEdg       USGraph

Theoremusgrfs 39405* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 39403. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgrfun 39406 The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USGraph iEdg

Theoremusgrusgra 39407 A simple graph represented by a class induces a representation as binary relation. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
USGraph Vtx USGrph iEdg

Theoremusgredgss 39408* The set of edges of a simple graph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
USGraph Edg Vtx

Theoremedgusgr 39409 An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
USGraph Edg Vtx

Theoremisusgrop 39410* The property of being an undirected simple graph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 30-Nov-2020.)
USGraph

Theoremusgrop 39411 A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.) (Proof shortened by AV, 30-Nov-2020.)
USGraph Vtx iEdg USGraph

Theoremisausgr 39412* The property of an unordered pair to be an alternatively defined simple graph, defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)

Theoremausgrusgrb 39413* The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)
USGraph

Theoremusgrausgri 39414* A simple graph represented by an alternatively defined simple graph. (Contributed by AV, 15-Oct-2020.)
USGraph VtxEdg

Theoremausgrumgri 39415* If an alternatively defined simple graph has the vertices and edges of an arbitrary graph, the arbitrary graph is an undirected multigraph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 25-Nov-2020.)
VtxEdg iEdg UMGraph

Theoremausgrusgri 39416* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 15-Oct-2020.)
VtxEdg iEdg USGraph

Theoremusgrausgrb 39417* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
iEdg VtxEdg USGraph

Theoremusgredgop 39418 An edge of a simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.) (Revised by AV, 15-Oct-2020.)
USGraph iEdg

Theoremusgrf1o 39419 The edge function of a simple graph is a bijective function onto its range. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
iEdg       USGraph

Theoremusgrf1 39420 The edge function of a simple graph is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
iEdg       USGraph

Theoremuspgrf1oedg 39421 The edge function of a simple graph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
iEdg       USPGraph Edg

Theoremusgrss 39422 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 15-Oct-2020.)
iEdg       Vtx       USGraph

Theoremuspgrushgr 39423 A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
USPGraph USHGraph

Theoremuspgrupgr 39424 A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
USPGraph UPGraph

Theoremuspgrupgrushgr 39425 A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.)
USPGraph UPGraph USHGraph

Theoremusgruspgr 39426 A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
USGraph USPGraph

Theoremusgrumgr 39427 A simple graph is an undirected multigraph. (Contributed by AV, 25-Nov-2020.)
USGraph UMGraph

Theoremusgrumgruspgr 39428 A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020.)
USGraph UMGraph USPGraph

Theoremusgruspgrb 39429* A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)
USGraph USPGraph Edg

Theoremusgrupgr 39430 A simple graph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 20-Aug-2017.) (Revised by AV, 15-Oct-2020.)
USGraph UPGraph

Theoremusgruhgr 39431 A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.)
USGraph UHGraph

Theoremusgrislfuspgr 39432* A simple graph is a loop-free simple pseudograph. (Contributed by AV, 27-Jan-2021.)
Vtx       iEdg       USGraph USPGraph

Theoremuspgrun 39433 The union of two simple pseudographs and with the same vertex set is a pseudograph with the vertex and the union of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)
USPGraph        USPGraph        iEdg       iEdg       Vtx       Vtx                                    Vtx        iEdg        UPGraph

Theoremuspgrunop 39434 The union of two simple pseudographs (with the same vertex set): If and are simple pseudographs, then is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
USPGraph        USPGraph        iEdg       iEdg       Vtx       Vtx                             UPGraph

Theoremusgrun 39435 The union of two simple graphs and with the same vertex set is a multigraph (not necessarily a simple graph!) with the vertex and the union of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
USGraph        USGraph        iEdg       iEdg       Vtx       Vtx                                    Vtx        iEdg        UMGraph

Theoremusgrunop 39436 The union of two simple graphs (with the same vertex set): If and are simple graphs, then is a multigraph (not necessarily a simple graph!) - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.)
USGraph        USGraph        iEdg       iEdg       Vtx       Vtx                             UMGraph

Theoremusgredg2 39437 The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
iEdg       USGraph

Theoremusgredg2ALT 39438 Alternate proof of usgredg2 39437, not using umgredg2 39345. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
iEdg       USGraph

Theoremusgredgprv 39439 In a simple graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
iEdg       Vtx       USGraph

TheoremusgredgprvALT 39440 Alternate proof of usgredgprv 39439, using usgredg2 39437 instead of umgredgprv 39352. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
iEdg       Vtx       USGraph

Theoremusgredgappr 39441 An edge of a simple graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge). Analogue of usgredg2 39437. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.)
Edg       USGraph

Theoremusgrpredgav 39442 An edge of a simple graph always connects two vertices. Analogue of usgredgprv 39439. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
Edg       Vtx       USGraph

Theoremedgassv2 39443 An edge of a simple 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.) (Revised by AV, 23-Oct-2020.)
Vtx       Edg       USGraph

Theoremusgredg 39444* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Shortened by AV, 25-Nov-2020.)
Vtx       Edg       USGraph

Theoremusgrnloopv 39445 In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
iEdg       USGraph

TheoremusgrnloopvALT 39446 Alternate proof of usgrnloopv 39445, not using umgrnloopv 39351. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
iEdg       USGraph

Theoremusgrnloop 39447* In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
iEdg       USGraph

TheoremusgrnloopALT 39448* Alternate proof of usgrnloop 39447, not using umgrnloop 39353. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
iEdg       USGraph

Theoremusgrnloop0 39449* A simple graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
iEdg       USGraph

Theoremusgrnloop0ALT 39450* Alternate proof of usgrnloop0 39449, not using umgrnloop0 39354. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
iEdg       USGraph

Theoremusgredgne 39451 An edge of a simple graph always connects two different vertices. Analogue of usgrnloopv 39445 resp. usgrnloop 39447. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
Edg       USGraph

Theoremusgrf1oedg 39452 The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by by AV, 18-Oct-2020.)
iEdg       Edg       USGraph

Theoremuhgr2edg 39453* If a vertex is adjacent to two different vertices in a hypergraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
iEdg       Edg       Vtx       UHGraph

Theoremumgr2edg 39454* If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
iEdg       Edg       UMGraph

Theoremusgr2edg 39455* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Feb-2021.)
iEdg       Edg       USGraph

Theoremusgr2edg1 39456* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.)
iEdg       Edg       USGraph

Theoremusgredg3 39457* The value of the "edge function" of a simple graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgredg4 39458* For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgredgreu 39459* For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgredg2vtx 39460* For a vertex incident to an edge there is another vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 5-Dec-2020.)
USGraph Edg Vtx

Theoremuspgredg2vtxeu 39461* For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 6-Dec-2020.)
USPGraph Edg Vtx

Theoremusgredg2vtxeu 39462* For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)
USGraph Edg Vtx

Theoremusgredg2vtxeuALT 39463* Alternate proof of usgredg2vtxeu 39462, using edgiedgb 39377, the general translation from iEdg to Edg. (Contributed by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
USGraph Edg Vtx

Theoremuspgredg2vlem 39464* Lemma for uspgredg2v 39465. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
Vtx       Edg              USPGraph

Theoremuspgredg2v 39465* In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
Vtx       Edg                     USPGraph

Theoremusgredg2vlem1 39466* Lemma 1 for usgredg2v 39468. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg              USGraph

Theoremusgredg2vlem2 39467* Lemma 2 for usgredg2v 39468. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg              USGraph

Theoremusgredg2v 39468* In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg                     USGraph

Theoremusgredgleord 39469* In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg       USGraph

Theoremushgredgedga 39470* In a simple hypergraph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 11-Dec-2020.)
Edg       iEdg       Vtx                            USHGraph

Theoremusgredgedga 39471* In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
Edg       iEdg       Vtx                            USGraph

Theoremushgredgedgaloop 39472* In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex and the set of loops at this vertex. (Contributed by AV, 11-Dec-2020.)
Edg       iEdg       Vtx                            USHGraph

Theoremuspgredgaleord 39473* In a simple pseudograph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
Vtx       Edg       USPGraph

Theoremusgredgaleord 39474* In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)
Vtx       Edg       USGraph

TheoremusgredgaleordALT 39475* In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Vtx       Edg       USGraph

21.33.8.8  Examples for graphs

Theoremusgr0e 39476 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.) (Proof shortened by AV, 25-Nov-2020.)
iEdg        USGraph

Theoremusgr0vb 39477 The null graph, with no vertices, is a simple graph iff the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 16-Oct-2020.)
Vtx USGraph iEdg

Theoremuhgr0v0e 39478 The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.)
Vtx       Edg       UHGraph

Theoremuhgr0vsize0 39479 The size of a hypergraph with 0 vertices (the null graph) is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 7-Nov-2020.)
Vtx       Edg       UHGraph

Theoremusgr0v 39480 The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.)
Vtx iEdg USGraph

Theoremuhgr0vusgr 39481 The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.)
UHGraph Vtx USGraph

Theoremusgr0 39482 The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
USGraph

Theoremuspgr1e 39483 A simple pseudograph with one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.)
Vtx                            iEdg        USPGraph

Theoremusgr1e 39484 A simple graph with one edge ( with additional assumption that since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.)
Vtx                            iEdg               USGraph

Theoremusgr0eop 39485 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.)
USGraph

Theoremuspgr1eop 39486 A simple pseudograph with (at least) two vertices and one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
USPGraph

Theoremuspgr1ewop 39487 A simple pseudograph with (at least) two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
USPGraph

Theoremuspgr1v1eop 39488 A simple pseudograph with (at least) one vertex and one edge (a loop). (Contributed by AV, 5-Dec-2020.)
USPGraph

Theoremusgr1eop 39489 A simple graph with (at least) two different vertices and one edge. If the two vertices were not different, the edge would be a loop. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.)
USGraph

Theoremuspgr2v1e2w 39490 A simple pseudograph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
USPGraph

Theoremusgr2v1e2w 39491 A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
USGraph

Theoremedg0usgr 39492 A class without edges is a simple graph. Since does not generally imply , but iEdg is required for to be a simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.)
Edg iEdg USGraph

Theoremusgr1vr 39493 A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.)
Vtx USGraph iEdg

Theoremusgr1v 39494 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.)
Vtx USGraph iEdg

Theoremusgr1v0edg 39495 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.)
Vtx iEdg USGraph Edg

Theoremusgrexmpllem 39496 Lemma for usgrexmpl 39499. (Contributed by AV, 21-Oct-2020.)
Vtx iEdg

Theoremusgrexmplvtx 39497 The vertices of the graph . (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.)
Vtx

Theoremusgrexmpledg 39498 The edges of the graph . (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.)
Edg

Theoremusgrexmpl 39499 is a simple graph of five vertices , with edges . (Contributed by Alexander van der Vekens, 15-Aug-2017.) (Revised by AV, 21-Oct-2020.)
USGraph

Theoremgriedg0prc 39500* The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.)

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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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