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Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremumgr0e 26901 The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ UMGraph)

Theoremupgr0e 26902 The empty graph, with vertices but no edges, is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ UPGraph)

Theoremupgr1elem 26903* Lemma for upgr1e 26904 and uspgr1e 27032. (Contributed by AV, 16-Oct-2020.)
(𝜑 → {𝐵, 𝐶} ∈ 𝑆)    &   (𝜑𝐵𝑊)       (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝑆 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Theoremupgr1e 26904 A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1e 27032. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})       (𝜑𝐺 ∈ UPGraph)

Theoremupgr0eop 26905 The empty graph, with vertices but no edges, is a pseudograph. The empty graph is actually a simple graph, see usgr0eop 27034, and therefore also a multigraph (𝐺 ∈ UMGraph). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.)
(𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Theoremupgr1eop 26906 A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1eop 27035. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph)

Theoremupgr0eopALT 26907 Alternate proof of upgr0eop 26905, using the general theorem gropeld 26824 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr0eop 26905). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Theoremupgr1eopALT 26908 Alternate proof of upgr1eop 26906, using the general theorem gropeld 26824 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr1eop 26906). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph)

Theoremupgrun 26909 The union 𝑈 of two pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ UPGraph)    &   (𝜑𝐻 ∈ UPGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UPGraph)

Theoremupgrunop 26910 The union of two pseudographs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are pseudographs, then 𝑉, 𝐸𝐹 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ UPGraph)    &   (𝜑𝐻 ∈ UPGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph)

Theoremumgrun 26911 The union 𝑈 of two multigraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a multigraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 25-Nov-2020.)
(𝜑𝐺 ∈ UMGraph)    &   (𝜑𝐻 ∈ UMGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UMGraph)

Theoremumgrunop 26912 The union of two multigraphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are multigraphs, then 𝑉, 𝐸𝐹 is a multigraph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
(𝜑𝐺 ∈ UMGraph)    &   (𝜑𝐻 ∈ UMGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UMGraph)

16.2.3  Loop-free graphs

For a hypergraph, the property to be "loop-free" is expressed by 𝐼:dom 𝐼𝐸 with 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} and 𝐼 = (iEdg‘𝐺). 𝐸 is the set of edges which connect at least two vertices.

Theoremumgrislfupgrlem 26913 Lemma for umgrislfupgr 26914 and usgrislfuspgr 26975. (Contributed by AV, 27-Jan-2021.)
({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}

Theoremumgrislfupgr 26914* A multigraph is a loop-free pseudograph. (Contributed by AV, 27-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UMGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))

Theoremlfgredgge2 26915* An edge of a loop-free graph has at least two ends. (Contributed by AV, 23-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}       ((𝐼:𝐴𝐸𝑋𝐴) → 2 ≤ (♯‘(𝐼𝑋)))

Theoremlfgrnloop 26916* A loop-free graph has no loops. (Contributed by AV, 23-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}       (𝐼:𝐴𝐸 → {𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}} = ∅)

16.2.4  Edges as subsets of vertices of graphs

Theoremuhgredgiedgb 26917* In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)
𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))

Theoremuhgriedg0edg0 26918 A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 8-Dec-2021.)
(𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))

Theoremuhgredgn0 26919 An edge of a hypergraph is a nonempty subset of vertices. (Contributed by AV, 28-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Theoremedguhgr 26920 An edge of a hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 28-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ 𝒫 (Vtx‘𝐺))

Theoremuhgredgrnv 26921 An edge of a hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 4-Jun-2021.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑁𝐸) → 𝑁 ∈ (Vtx‘𝐺))

Theoremuhgredgss 26922 The set of edges of a hypergraph is a subset of the power set of vertices without the empty set. (Contributed by AV, 29-Nov-2020.)
(𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Theoremupgredgss 26923* The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 29-Nov-2020.)
(𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Theoremumgredgss 26924* The set of edges of a multigraph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 25-Nov-2020.)
(𝐺 ∈ UMGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})

Theoremedgupgr 26925 Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)
((𝐺 ∈ UPGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2))

Theoremedgumgr 26926 Properties of an edge of a multigraph. (Contributed by AV, 25-Nov-2020.)
((𝐺 ∈ UMGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐸) = 2))

Theoremuhgrvtxedgiedgb 26927* In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 6-Jul-2022.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖) ↔ ∃𝑒𝐸 𝑈𝑒))

Theoremupgredg 26928* For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.) (Proof shortened by AV, 26-Nov-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})

Theoremumgredg 26929* For each edge in a multigraph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 25-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝐶 = {𝑎, 𝑏}))

Theoremupgrpredgv 26930 An edge of a pseudograph always connects two vertices if the edge contains two sets. The two vertices/sets need not necessarily be different (loops are allowed). (Contributed by AV, 18-Nov-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ (𝑀𝑈𝑁𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀𝑉𝑁𝑉))

Theoremumgrpredgv 26931 An edge of a multigraph always connects two vertices. Analogue of umgredgprv 26898. This theorem does not hold for arbitrary pseudographs: if either 𝑀 or 𝑁 is a proper class, then {𝑀, 𝑁} ∈ 𝐸 could still hold ({𝑀, 𝑁} would be either {𝑀} or {𝑁}, see prprc1 4675 or prprc2 4676, i.e. a loop), but 𝑀𝑉 or 𝑁𝑉 would not be true. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀𝑉𝑁𝑉))

Theoremupgredg2vtx 26932* For a vertex incident to an edge there is another vertex incident to the edge in a pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 5-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐶𝐸𝐴𝐶) → ∃𝑏𝑉 𝐶 = {𝐴, 𝑏})

Theoremupgredgpr 26933 If a proper pair (of vertices) is a subset of an edge in a pseudograph, the pair is the edge. (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UPGraph ∧ 𝐶𝐸 ∧ {𝐴, 𝐵} ⊆ 𝐶) ∧ (𝐴𝑈𝐵𝑊𝐴𝐵)) → {𝐴, 𝐵} = 𝐶)

Theoremedglnl 26934* The edges incident with a vertex 𝑁 are the edges joining 𝑁 with other vertices and the loops on 𝑁 in a pseudograph. (Contributed by AV, 18-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})

Theoremnumedglnl 26935* The number of edges incident with a vertex 𝑁 is the number of edges joining 𝑁 with other vertices and the number of loops on 𝑁 in a pseudograph of finite size. (Contributed by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))

Theoremumgredgne 26936 An edge of a multigraph always connects two different vertices. Analogue of umgrnloopv 26897 resp. umgrnloop 26899. (Contributed by AV, 27-Nov-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → 𝑀𝑁)

Theoremumgrnloop2 26937 A multigraph has no loops. (Contributed by AV, 27-Oct-2020.) (Revised by AV, 30-Nov-2020.)
(𝐺 ∈ UMGraph → {𝑁, 𝑁} ∉ (Edg‘𝐺))

Theoremumgredgnlp 26938* An edge of a multigraph is not a loop. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝐶𝐸) → ¬ ∃𝑣 𝐶 = {𝑣})

16.2.5  Undirected simple graphs

In this section, "simple graph" will always stand for "undirected simple graph (without loops)" and "simple pseudograph" for "undirected simple pseudograph (which could have loops)".

Syntaxcuspgr 26939 Extend class notation with undirected simple pseudographs (which could have loops).
class USPGraph

Syntaxcusgr 26940 Extend class notation with undirected simple graphs (without loops).
class USGraph

Definitiondf-uspgr 26941* Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph (see uspgrupgr 26967) or a special undirected simple hypergraph (see uspgrushgr 26966), consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}

Definitiondf-usgr 26942* Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph (see usgruspgr 26969), consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to an undirected simple pseudograph, an undirected simple graph has no loops (edges connecting a vertex with itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}

Theoremisuspgr 26943* The property of being a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))

Theoremisusgr 26944* The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))

Theoremuspgrf 26945* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Theoremusgrf 26946* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})

Theoremisusgrs 26947* The property of being a simple graph, simplified version of isusgr 26944. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.) (Proof shortened by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))

Theoremusgrfs 26948* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 26946. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})

Theoremusgrfun 26949 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.)
(𝐺 ∈ USGraph → Fun (iEdg‘𝐺))

Theoremusgredgss 26950* 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.)
(𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})

Theoremedgusgr 26951 An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐸) = 2))

Theoremisuspgrop 26952* The property of being an undirected simple pseudograph 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, 25-Nov-2021.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))

Theoremisusgrop 26953* 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.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}))

Theoremusgrop 26954 A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.) (Proof shortened by AV, 30-Nov-2020.)
(𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph)

Theoremisausgr 26955* 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.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}       ((𝑉𝑊𝐸𝑋) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))

Theoremausgrusgrb 26956* The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}       ((𝑉𝑋𝐸𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph))

Theoremusgrausgri 26957* A simple graph represented by an alternatively defined simple graph. (Contributed by AV, 15-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}       (𝐻 ∈ USGraph → (Vtx‘𝐻)𝐺(Edg‘𝐻))

Theoremausgrumgri 26958* 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.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}       ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → 𝐻 ∈ UMGraph)

Theoremausgrusgri 26959* 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.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}    &   𝑂 = {𝑓𝑓:dom 𝑓1-1→ran 𝑓}       ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph)

Theoremusgrausgrb 26960* 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.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}    &   𝑂 = {𝑓𝑓:dom 𝑓1-1→ran 𝑓}       ((𝐻𝑊 ∧ (iEdg‘𝐻) ∈ 𝑂) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ 𝐻 ∈ USGraph))

Theoremusgredgop 26961 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.)
((𝐺 ∈ USGraph ∧ 𝐸 = (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} ↔ ⟨𝑋, {𝑀, 𝑁}⟩ ∈ 𝐸))

Theoremusgrf1o 26962 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.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1-onto→ran 𝐸)

Theoremusgrf1 26963 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.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→ran 𝐸)

Theoremuspgrf1oedg 26964 The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))

Theoremusgrss 26965 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸𝑋) ⊆ 𝑉)

Theoremuspgrushgr 26966 A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Theoremuspgrupgr 26967 A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)

Theoremuspgrupgrushgr 26968 A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.)
(𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))

Theoremusgruspgr 26969 A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)

Theoremusgrumgr 26970 A simple graph is an undirected multigraph. (Contributed by AV, 25-Nov-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)

Theoremusgrumgruspgr 26971 A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020.)
(𝐺 ∈ USGraph ↔ (𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph))

Theoremusgruspgrb 26972* A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)
(𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))

Theoremusgrupgr 26973 A simple graph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 20-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)

Theoremusgruhgr 26974 A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)

Theoremusgrislfuspgr 26975* A simple graph is a loop-free simple pseudograph. (Contributed by AV, 27-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))

Theoremuspgrun 26976 The union 𝑈 of two simple pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)
(𝜑𝐺 ∈ USPGraph)    &   (𝜑𝐻 ∈ USPGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UPGraph)

Theoremuspgrunop 26977 The union of two simple pseudographs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are simple pseudographs, then 𝑉, 𝐸𝐹 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.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ USPGraph)    &   (𝜑𝐻 ∈ USPGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph)

Theoremusgrun 26978 The union 𝑈 of two simple graphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a multigraph (not necessarily a simple graph!) with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
(𝜑𝐺 ∈ USGraph)    &   (𝜑𝐻 ∈ USGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UMGraph)

Theoremusgrunop 26979 The union of two simple graphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are simple graphs, then 𝑉, 𝐸𝐹 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.)
(𝜑𝐺 ∈ USGraph)    &   (𝜑𝐻 ∈ USGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UMGraph)

Theoremusgredg2 26980 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.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸𝑋)) = 2)

Theoremusgredg2ALT 26981 Alternate proof of usgredg2 26980, not using umgredg2 26891. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸𝑋)) = 2)

Theoremusgredgprv 26982 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.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} → (𝑀𝑉𝑁𝑉)))

TheoremusgredgprvALT 26983 Alternate proof of usgredgprv 26982, using usgredg2 26980 instead of umgredgprv 26898. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} → (𝑀𝑉𝑁𝑉)))

Theoremusgredgppr 26984 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 26980. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → (♯‘𝐶) = 2)

Theoremusgrpredgv 26985 An edge of a simple graph always connects two vertices. Analogue of usgredgprv 26982. (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.)
𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀𝑉𝑁𝑉))

Theoremedgssv2 26986 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → (𝐶𝑉 ∧ (♯‘𝐶) = 2))

Theoremusgredg 26987* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝐶 = {𝑎, 𝑏}))

Theoremusgrnloopv 26988 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.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀𝑊) → ((𝐸𝑋) = {𝑀, 𝑁} → 𝑀𝑁))

TheoremusgrnloopvALT 26989 Alternate proof of usgrnloopv 26988, not using umgrnloopv 26897. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀𝑊) → ((𝐸𝑋) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremusgrnloop 26990* 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.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑀, 𝑁} → 𝑀𝑁))

TheoremusgrnloopALT 26991* Alternate proof of usgrnloop 26990, not using umgrnloop 26899. (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.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremusgrnloop0 26992* 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.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) = {𝑈}} = ∅)

Theoremusgrnloop0ALT 26993* Alternate proof of usgrnloop0 26992, not using umgrnloop0 26900. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) = {𝑈}} = ∅)

Theoremusgredgne 26994 An edge of a simple graph always connects two different vertices. Analogue of usgrnloopv 26988 resp. usgrnloop 26990. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → 𝑀𝑁)

Theoremusgrf1oedg 26995 The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by AV, 18-Oct-2020.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1-onto𝐸)

Theoremuhgr2edg 26996* 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.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremumgr2edg 26997* 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.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremusgr2edg 26998* 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.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremumgr2edg1 26999* If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 8-Jun-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼𝑥))

Theoremusgr2edg1 27000* 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.) (Proof shortened by AV, 8-Jun-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼𝑥))

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