P. 293 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	usgrausgri 29201*	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))
Theorem	ausgrumgri 29202*	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}}    ⇒   ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → 𝐻 ∈ UMGraph)
Theorem	ausgrusgri 29203*	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 = {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓}    ⇒   ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph)
Theorem	usgrausgrb 29204*	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 = {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓}    ⇒   ⊢ ((𝐻 ∈ 𝑊 ∧ (iEdg‘𝐻) ∈ 𝑂) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ 𝐻 ∈ USGraph))
Theorem	usgredgop 29205	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) → ((𝐸‘𝑋) = {𝑀, 𝑁} ↔ ⟨𝑋, {𝑀, 𝑁}⟩ ∈ 𝐸))
Theorem	usgrf1o 29206	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 𝐸–1-1-onto→ran 𝐸)
Theorem	usgrf1 29207	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 𝐸–1-1→ran 𝐸)
Theorem	uspgrf1oedg 29208	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 𝐸–1-1-onto→(Edg‘𝐺))
Theorem	usgrss 29209	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) → (𝐸‘𝑋) ⊆ 𝑉)
Theorem	uspgredgiedg 29210*	In a simple pseudograph, for each edge there is exactly one indexed edge. (Contributed by AV, 20-Apr-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼 𝐾 = (𝐼‘𝑥))
Theorem	uspgriedgedg 29211*	In a simple pseudograph, for each indexed edge there is exactly one edge. (Contributed by AV, 20-Apr-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋))
Theorem	uspgrushgr 29212	A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
Theorem	uspgrupgr 29213	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)
Theorem	uspgrupgrushgr 29214	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))
Theorem	usgruspgr 29215	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)
Theorem	usgrumgr 29216	A simple graph is an undirected multigraph. (Contributed by AV, 25-Nov-2020.)
⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
Theorem	usgrumgruspgr 29217	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))
Theorem	usgruspgrb 29218*	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))
Theorem	uspgruhgr 29219	An undirected simple pseudograph is an undirected hypergraph. (Contributed by AV, 21-Apr-2025.)
⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
Theorem	usgrupgr 29220	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)
Theorem	usgruhgr 29221	A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.)
⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
Theorem	usgrislfuspgr 29222*	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)}))
Theorem	uspgrun 29223	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 (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)
⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))    ⇒   ⊢ (𝜑 → 𝑈 ∈ UPGraph)
Theorem	uspgrunop 29224	The union of two simple pseudographs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are simple pseudographs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ 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.)
⊢ (𝜑 → 𝐺 ∈ USPGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USPGraph)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    ⇒   ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)
Theorem	usgrun 29225	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 (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
⊢ (𝜑 → 𝐺 ∈ USGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USGraph)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))    ⇒   ⊢ (𝜑 → 𝑈 ∈ UMGraph)
Theorem	usgrunop 29226	The union of two simple graphs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are simple graphs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ 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.)
⊢ (𝜑 → 𝐺 ∈ USGraph)    &   ⊢ (𝜑 → 𝐻 ∈ USGraph)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    ⇒   ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UMGraph)
Theorem	usgredg2 29227	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) → (♯‘(𝐸‘𝑋)) = 2)
Theorem	usgredg2ALT 29228	Alternate proof of usgredg2 29227, not using umgredg2 29135. (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) → (♯‘(𝐸‘𝑋)) = 2)
Theorem	usgredgprv 29229	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) → ((𝐸‘𝑋) = {𝑀, 𝑁} → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
Theorem	usgredgprvALT 29230	Alternate proof of usgredgprv 29229, using usgredg2 29227 instead of umgredgprv 29142. (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) → ((𝐸‘𝑋) = {𝑀, 𝑁} → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)))
Theorem	usgredgppr 29231	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 29227. (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 ∧ 𝐶 ∈ 𝐸) → (♯‘𝐶) = 2)
Theorem	usgrpredgv 29232	An edge of a simple graph always connects two vertices. Analogue of usgredgprv 29229. (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) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
Theorem	edgssv2 29233	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 ∧ 𝐶 ∈ 𝐸) → (𝐶 ⊆ 𝑉 ∧ (♯‘𝐶) = 2))
Theorem	usgredg 29234*	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 ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝐶 = {𝑎, 𝑏}))
Theorem	usgrnloopv 29235	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 ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁))
Theorem	usgrnloopvALT 29236	Alternate proof of usgrnloopv 29235, not using umgrnloopv 29141. (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 ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁))
Theorem	usgrnloop 29237*	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(𝐸‘𝑥) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁))
Theorem	usgrnloopALT 29238*	Alternate proof of usgrnloop 29237, not using umgrnloop 29143. (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(𝐸‘𝑥) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁))
Theorem	usgrnloop0 29239*	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 ∣ (𝐸‘𝑥) = {𝑈}} = ∅)
Theorem	usgrnloop0ALT 29240*	Alternate proof of usgrnloop0 29239, not using umgrnloop0 29144. (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 ∣ (𝐸‘𝑥) = {𝑈}} = ∅)
Theorem	usgredgne 29241	An edge of a simple graph always connects two different vertices. Analogue of usgrnloopv 29235 resp. usgrnloop 29237. (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) → 𝑀 ≠ 𝑁)
Theorem	usgrf1oedg 29242	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 𝐼–1-1-onto→𝐸)
Theorem	uhgr2edg 29243*	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 ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)))
Theorem	umgr2edg 29244*	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 ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)))
Theorem	usgr2edg 29245*	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 ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼∃𝑦 ∈ dom 𝐼(𝑥 ≠ 𝑦 ∧ 𝑁 ∈ (𝐼‘𝑥) ∧ 𝑁 ∈ (𝐼‘𝑦)))
Theorem	umgr2edg1 29246*	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 ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥))
Theorem	usgr2edg1 29247*	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 ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼‘𝑥))
Theorem	umgrvad2edg 29248*	If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex, analogous to usgr2edg 29245. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐸 (𝑥 ≠ 𝑦 ∧ 𝑁 ∈ 𝑥 ∧ 𝑁 ∈ 𝑦))
Theorem	umgr2edgneu 29249*	If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex, analogous to usgr2edg1 29247. Lemma for theorems about friendship graphs. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ≠ 𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ 𝐸 𝑁 ∈ 𝑥)
Theorem	usgrsizedg 29250	In a simple graph, the size of the edge function is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 7-Jun-2021.)
⊢ (𝐺 ∈ USGraph → (♯‘(iEdg‘𝐺)) = (♯‘(Edg‘𝐺)))
Theorem	usgredg3 29251*	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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ (𝐸‘𝑋) = {𝑥, 𝑦}))
Theorem	usgredg4 29252*	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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})
Theorem	usgredgreu 29253*	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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃!𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})
Theorem	usgredg2vtx 29254*	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.)
⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})
Theorem	uspgredg2vtxeu 29255*	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.)
⊢ ((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})
Theorem	usgredg2vtxeu 29256*	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.)
⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})
Theorem	usgredg2vtxeuALT 29257*	Alternate proof of usgredg2vtxeu 29256, using edgiedgb 29089, the general translation from (iEdg‘𝐺) to (Edg‘𝐺). (Contributed by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})
Theorem	uspgredg2vlem 29258*	Lemma for uspgredg2v 29259. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐴 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧}) ∈ 𝑉)
Theorem	uspgredg2v 29259*	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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐴 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}    &   ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)
Theorem	usgredg2vlem1 29260*	Lemma 1 for usgredg2v 29262. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
Theorem	usgredg2vlem2 29261*	Lemma 2 for usgredg2v 29262. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁}))
Theorem	usgredg2v 29262*	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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}    &   ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}))    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)
Theorem	usgriedgleord 29263*	Alternate version of usgredgleord 29268, not using the notation (Edg‘𝐺). 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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) ≤ (♯‘𝑉))
Theorem	ushgredgedg 29264*	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.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}    &   ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))    ⇒   ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)
Theorem	usgredgedg 29265*	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 AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}    &   ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)
Theorem	ushgredgedgloop 29266*	In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex 𝑁 and the set of loops at this vertex 𝑁. (Contributed by AV, 11-Dec-2020.) (Revised by AV, 6-Jul-2022.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}    &   ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑁}}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))    ⇒   ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)
Theorem	uspgredgleord 29267*	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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉))
Theorem	usgredgleord 29268*	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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉))
Theorem	usgredgleordALT 29269*	Alternate proof for usgredgleord 29268 based on usgriedgleord 29263. 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, 5-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉))
Theorem	usgrstrrepe 29270*	Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.)
⊢ 𝑉 = (Base‘𝐺)    &   ⊢ 𝐼 = (.ef‘ndx)    &   ⊢ (𝜑 → 𝐺 Struct 𝑋)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})    ⇒   ⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph)
17.2.6  Examples for graphs
Theorem	usgr0e 29271	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.)
⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → (iEdg‘𝐺) = ∅)    ⇒   ⊢ (𝜑 → 𝐺 ∈ USGraph)
Theorem	usgr0vb 29272	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.)
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))
Theorem	uhgr0v0e 29273	The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)
Theorem	uhgr0vsize0 29274	The size of a hypergraph with no vertices (the null graph) is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 7-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = 0)
Theorem	uhgr0edgfi 29275	A graph of order 0 (i.e. with 0 vertices) has a finite set of edges. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 8-Jun-2021.)
⊢ ((𝐺 ∈ UHGraph ∧ (♯‘(Vtx‘𝐺)) = 0) → (Edg‘𝐺) ∈ Fin)
Theorem	usgr0v 29276	The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.)
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph)
Theorem	uhgr0vusgr 29277	The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.)
⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph)
Theorem	usgr0 29278	The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
⊢ ∅ ∈ USGraph
Theorem	uspgr1e 29279	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.) (Proof shortened by AV, 17-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})    ⇒   ⊢ (𝜑 → 𝐺 ∈ USPGraph)
Theorem	usgr1e 29280	A 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 = (Vtx‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐺 ∈ USGraph)
Theorem	usgr0eop 29281	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 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)
Theorem	uspgr1eop 29282	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.)
⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ USPGraph)
Theorem	uspgr1ewop 29283	A simple pseudograph with (at least) two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ⟨𝑉, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USPGraph)
Theorem	uspgr1v1eop 29284	A simple pseudograph with (at least) one vertex and one edge (a loop). (Contributed by AV, 5-Dec-2020.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑉) → ⟨𝑉, {⟨𝐴, {𝐵}⟩}⟩ ∈ USPGraph)
Theorem	usgr1eop 29285	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.)
⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ≠ 𝐶 → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ USGraph))
Theorem	uspgr2v1e2w 29286	A simple pseudograph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ⟨{𝐴, 𝐵}, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USPGraph)
Theorem	usgr2v1e2w 29287	A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → ⟨{𝐴, 𝐵}, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USGraph)
Theorem	edg0usgr 29288	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 simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.)
⊢ ((𝐺 ∈ 𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)
Theorem	lfuhgr1v0e 29289*	A loop-free hypergraph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 2-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (Edg‘𝐺) = ∅)
Theorem	usgr1vr 29290	A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 2-Apr-2021.)
⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅))
Theorem	usgr1v 29291	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 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))
Theorem	usgr1v0edg 29292	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 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴} ∧ Fun (iEdg‘𝐺)) → (𝐺 ∈ USGraph ↔ (Edg‘𝐺) = ∅))
Theorem	usgrexmpldifpr 29293	Lemma for usgrexmpledg 29297: all "edges" are different. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
⊢ (({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3}))
Theorem	usgrexmplef 29294*	Lemma for usgrexmpl 29298. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
⊢ 𝑉 = (0...4)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    ⇒   ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
Theorem	usgrexmpllem 29295	Lemma for usgrexmpl 29298. (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)
Theorem	usgrexmplvtx 29296	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} ∪ {3, 4})
Theorem	usgrexmpledg 29297	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}} ∪ {{2, 0}, {0, 3}})
Theorem	usgrexmpl 29298	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.) (Proof shortened by AV, 7-Aug-2025.)
⊢ 𝑉 = (0...4)    &   ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   ⊢ 𝐺 = ⟨𝑉, 𝐸⟩    ⇒   ⊢ 𝐺 ∈ USGraph
Theorem	griedg0prc 29299*	The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}    ⇒   ⊢ 𝑈 ∉ V
Theorem	griedg0ssusgr 29300*	The class of all simple graphs is a superclass of the class of empty graphs represented as ordered pairs. (Contributed by AV, 27-Dec-2020.)
⊢ 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}    ⇒   ⊢ 𝑈 ⊆ USGraph