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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgriedg0prc 26201* 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

Theoremgriedg0ssusgr 26202* 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

Theoremusgrprc 26203 The class of simple graphs is a proper class (and therefore, because of prcssprc 4839, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.)
USGraph ∉ V

16.2.7  Subgraphs

Syntaxcsubgr 26204 Extend class notation with subgraphs.
class SubGraph

Definitiondf-subgr 26205* Define the class of the subgraph relation. A class 𝑠 is a subgraph of a class 𝑔 (the supergraph of 𝑠) if its vertices are also vertices of 𝑔, and its edges are also edges of 𝑔, connecting vertices of 𝑠 only (see section I.1 in [Bollobas] p. 2 or section 1.1 in [Diestel] p. 4). The second condition is ensured by the requirement that the edge function of 𝑠 is a restriction of the edge function of 𝑔 having only vertices of 𝑠 in its range. Note that the domains of the edge functions of the subgraph and the supergraph should be compatible. (Contributed by AV, 16-Nov-2020.)
SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}

Theoremrelsubgr 26206 The class of the subgraph relation is a relation. (Contributed by AV, 16-Nov-2020.)
Rel SubGraph

Theoremsubgrv 26207 If a class is a subgraph of another class, both classes are sets. (Contributed by AV, 16-Nov-2020.)
(𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))

Theoremissubgr 26208 The property of a set to be a subgraph of another set. (Contributed by AV, 16-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝑆)       ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))

Theoremissubgr2 26209 The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝑆)       ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))

Theoremsubgrprop 26210 The properties of a subgraph. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝑆)       (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))

Theoremsubgrprop2 26211 The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺, connecting vertices of the subgraph only. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝑆)       (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉))

Theoremuhgrissubgr 26212 The property of a hypergraph to be a subgraph. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝑆)    &   𝐵 = (iEdg‘𝐺)       ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵)))

Theoremsubgrprop3 26213 The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐴 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝑆)    &   𝐵 = (Edg‘𝐺)       (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))

Theoremegrsubgr 26214 An empty graph consisting of a subset of vertices of a graph (and having no edges) is a subgraph of the graph. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 17-Dec-2020.)
(((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)

Theorem0grsubgr 26215 The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
(𝐺𝑊 → ∅ SubGraph 𝐺)

Theorem0uhgrsubgr 26216 The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.)
((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)

Theoremuhgrsubgrself 26217 A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
(𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)

Theoremsubgrfun 26218 The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020.)
((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Theoremsubgruhgrfun 26219 The edge function of a subgraph of a hypergraph is a function. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 20-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Theoremsubgreldmiedg 26220 An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.)
((𝑆 SubGraph 𝐺𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))

Theoremsubgruhgredgd 26221 An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐼 = (iEdg‘𝑆)    &   (𝜑𝐺 ∈ UHGraph)    &   (𝜑𝑆 SubGraph 𝐺)    &   (𝜑𝑋 ∈ dom 𝐼)       (𝜑 → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))

Theoremsubumgredg2 26222* An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐼 = (iEdg‘𝑆)       ((𝑆 SubGraph 𝐺𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})

Theoremsubuhgr 26223 A subgraph of a hypergraph is a hypergraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph)

Theoremsubupgr 26224 A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)

Theoremsubumgr 26225 A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.)
((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UMGraph)

Theoremsubusgr 26226 A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 27-Nov-2020.)
((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph)

Theoremuhgrspansubgrlem 26227 Lemma for uhgrspansubgr 26228: The edges of the graph 𝑆 obtained by removing some edges of a hypergraph 𝐺 are subsets of its vertices (a spanning subgraph, see comment for uhgrspansubgr 26228. (Contributed by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph)       (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))

Theoremuhgrspansubgr 26228 A spanning subgraph 𝑆 of a hypergraph 𝐺 is actually a subgraph of 𝐺. A subgraph 𝑆 of a graph 𝐺 which has the same vertices as 𝐺 and is obtained by removing some edges of 𝐺 is called a spanning subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). Formally, the edges are "removed" by restricting the edge function of the original graph by an arbitrary class (which actually needs not to be a subset of the domain of the edge function). (Contributed by AV, 18-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph)       (𝜑𝑆 SubGraph 𝐺)

Theoremuhgrspan 26229 A spanning subgraph 𝑆 of a hypergraph 𝐺 is a hypergraph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph)       (𝜑𝑆 ∈ UHGraph)

Theoremupgrspan 26230 A spanning subgraph 𝑆 of a pseudograph 𝐺 is a pseudograph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UPGraph)       (𝜑𝑆 ∈ UPGraph)

Theoremumgrspan 26231 A spanning subgraph 𝑆 of a multigraph 𝐺 is a multigraph. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UMGraph)       (𝜑𝑆 ∈ UMGraph)

Theoremusgrspan 26232 A spanning subgraph 𝑆 of a simple graph 𝐺 is a simple graph. (Contributed by AV, 15-Oct-2020.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ USGraph)       (𝜑𝑆 ∈ USGraph)

Theoremuhgrspanop 26233 A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UHGraph)

Theoremupgrspanop 26234 A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UPGraph)

Theoremumgrspanop 26235 A spanning subgraph of a multigraph represented by an ordered pair is a multigraph. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UMGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UMGraph)

Theoremusgrspanop 26236 A spanning subgraph of a simple graph represented by an ordered pair is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ USGraph)

Theoremuhgrspan1lem1 26237 Lemma 1 for uhgrspan1 26240. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}       ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)

Theoremuhgrspan1lem2 26238 Lemma 2 for uhgrspan1 26240. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       (Vtx‘𝑆) = (𝑉 ∖ {𝑁})

Theoremuhgrspan1lem3 26239 Lemma 3 for uhgrspan1 26240. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       (iEdg‘𝑆) = (𝐼𝐹)

Theoremuhgrspan1 26240* The induced subgraph 𝑆 of a hypergraph 𝐺 obtained by removing one vertex is actually a subgraph of 𝐺. A subgraph is called induced or spanned by a subset of vertices of a graph if it contains all edges of the original graph that join two vertices of the subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → 𝑆 SubGraph 𝐺)

Theoremupgrreslem 26241* Lemma for upgrres 26243. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})

Theoremumgrreslem 26242* Lemma for umgrres 26244 and usgrres 26245. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})

Theoremupgrres 26243* A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 26240) is a pseudograph. (Contributed by AV, 8-Nov-2020.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)

Theoremumgrres 26244* A subgraph obtained by removing one vertex and all edges incident with this vertex from a multigraph (see uhgrspan1 26240) is a multigraph. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → 𝑆 ∈ UMGraph)

Theoremusgrres 26245* A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 26240) is a simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph)

Theoremupgrres1lem1 26246* Lemma 1 for upgrres1 26250. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)

Theoremumgrres1lem 26247* Lemma for umgrres1 26251. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})

Theoremupgrres1lem2 26248* Lemma 2 for upgrres1 26250. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (Vtx‘𝑆) = (𝑉 ∖ {𝑁})

Theoremupgrres1lem3 26249* Lemma 3 for upgrres1 26250. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (iEdg‘𝑆) = ( I ↾ 𝐹)

Theoremupgrres1 26250* A pseudograph obtained by removing one vertex and all edges incident with this vertex is a pseudograph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 26205 since the domains of the edge functions may not be compatible. (Contributed by AV, 8-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)

Theoremumgrres1 26251* A multigraph obtained by removing one vertex and all edges incident with this vertex is a multigraph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 26205 since the domains of the edge functions may not be compatible. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → 𝑆 ∈ UMGraph)

Theoremusgrres1 26252* Restricting a simple graph by removing one vertex results in a simple graph. Remark: This restricted graph is not a subgraph of the original graph in the sense of df-subgr 26205 since the domains of the edge functions may not be compatible. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph)

16.2.8  Finite undirected simple graphs

Syntaxcfusgr 26253 Extend class notation with finite simple graphs.
class FinUSGraph

Definitiondf-fusgr 26254 Define the class of all finite undirected simple graphs without loops (called "finite simple graphs" in the following). A finite simple graph is an undirected simple graph of finite order, i.e. with a finite set of vertices. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}

Theoremisfusgr 26255 The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))

Theoremfusgrvtxfi 26256 A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Theoremisfusgrf1 26257* The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (𝐺 ∈ FinUSGraph ↔ (𝐼:dom 𝐼1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ 𝑉 ∈ Fin)))

Theoremisfusgrcl 26258 The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 9-Jan-2020.)
(𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (#‘(Vtx‘𝐺)) ∈ ℕ0))

Theoremfusgrusgr 26259 A finite simple graph is a simple graph. (Contributed by AV, 16-Jan-2020.) (Revised by AV, 21-Oct-2020.)
(𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)

Theoremopfusgr 26260 A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.)
((𝑉𝑋𝐸𝑌) → (⟨𝑉, 𝐸⟩ ∈ FinUSGraph ↔ (⟨𝑉, 𝐸⟩ ∈ USGraph ∧ 𝑉 ∈ Fin)))

Theoremusgredgffibi 26261 The number of edges in a simple graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 22-Oct-2020.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (𝐸 ∈ Fin ↔ 𝐼 ∈ Fin))

Theoremfusgredgfi 26262* In a finite simple graph the number of edges which contain a given vertex is also finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 21-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ∈ Fin)

Theoremusgr1v0e 26263 The size of a (finite) simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 22-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ (#‘𝑉) = 1) → (#‘𝐸) = 0)

Theoremusgrfilem 26264* In a finite simple graph, the number of edges is finite iff the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin))

Theoremfusgrfisbase 26265 Induction base for fusgrfis 26267. Main work is done in uhgr0v0e 26175. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.)
(((𝑉𝑋𝐸𝑌) ∧ ⟨𝑉, 𝐸⟩ ∈ USGraph ∧ (#‘𝑉) = 0) → 𝐸 ∈ Fin)

Theoremfusgrfisstep 26266* Induction step in fusgrfis 26267: In a finite simple graph, the number of edges is finite if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.)
(((𝑉𝑋𝐸𝑌) ∧ ⟨𝑉, 𝐸⟩ ∈ FinUSGraph ∧ 𝑁𝑉) → (( I ↾ {𝑝 ∈ (Edg‘⟨𝑉, 𝐸⟩) ∣ 𝑁𝑝}) ∈ Fin → 𝐸 ∈ Fin))

Theoremfusgrfis 26267 A finite simple graph is of finite size, i.e. has a finite number of edges. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 8-Nov-2020.)
(𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)

Theoremfusgrfupgrfs 26268 A finite simple graph is a finite pseudograph of finite size. (Contributed by AV, 27-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ FinUSGraph → (𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin))

16.2.9  Neighbors, complete graphs and universal vertices

16.2.9.1  Neighbors

Syntaxcnbgr 26269 Extend class notation with neighbors (of a vertex in a graph).
class NeighbVtx

Definitiondf-nbgr 26270* Define the (open) neighborhood resp. the class of all neighbors of a vertex (in a graph), see definition in section I.1 of [Bollobas] p. 3 or definition in section 1.1 of [Diestel] p. 3. The neighborhood/neighbors of a vertex are all (other) vertices which are connected with this vertex by an edge. In contrast to a closed neighborhood, a vertex is not a neighbor of itself. This definition is applicable even for arbitrary hypergraphs.

Remark: To distinguish this definition from other definitions for neighborhoods resp. neighbors (e.g., nei in Topology, see df-nei 20950), the suffix Vtx is added to the class constant NeighbVtx. (Contributed by Alexander van der Vekens and Mario Carneiro, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.)

NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})

Theoremnbgrprc0 26271 The set of neighbors is empty if the graph 𝐺 or the vertex 𝑁 are proper classes. (Contributed by AV, 26-Oct-2020.)
(¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)

Theoremnbgrcl 26272 If a class 𝑋 has at least one neighbor, this class must be a vertex. (Contributed by AV, 6-Jun-2021.) (Revised by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋𝑉)

TheoremnbgrclOLD 26273 Obsolete version of nbgrcl 26272 as of 12-Feb-2022. (Contributed by AV, 6-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))

Theoremnbgrval 26274* The set of neighbors of a vertex 𝑉 in a graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})

Theoremdfnbgr2 26275* Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})

Theoremdfnbgr3 26276* Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 26274). (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})

Theoremnbgrnvtx0 26277 If a class 𝑋 is not a vertex of a graph 𝐺, then it has no neighbors in 𝐺. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝑋𝑉 → (𝐺 NeighbVtx 𝑋) = ∅)

Theoremnbgrel 26278* Characterization of a neighbor 𝑁 of a vertex 𝑋 in a graph 𝐺. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))

TheoremnbgrelOLD 26279* Obsolete version of nbgrel 26278 as of 12-Feb-2022. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑊 → (𝐾 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝐾𝑉𝑁𝑉) ∧ 𝐾𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝐾} ⊆ 𝑒)))

Theoremnbgrisvtx 26280 Every neighbor 𝑁 of a vertex 𝐾 is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁𝑉)

Theoremnbgrssvtx 26281 The neighbors of a vertex 𝐾 in a graph form a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝐺 NeighbVtx 𝐾) ⊆ 𝑉

TheoremnbgrisvtxOLD 26282 Obsolete version of nbgrisvtx 26280 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁𝑉)

TheoremnbgrssvtxOLD 26283 Obsolete version of nbgrssvtx 26281 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑉)

Theoremnbuhgr 26284* The set of neighbors of a vertex in a hypergraph. This version of nbgrval 26274 (with 𝑁 being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})

Theoremnbupgr 26285* The set of neighbors of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸})

Theoremnbupgrel 26286 A neighbor of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UPGraph ∧ 𝐾𝑉) ∧ (𝑁𝑉𝑁𝐾)) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸))

Theoremnbumgrvtx 26287* The set of neighbors of a vertex in a multigraph. (Contributed by AV, 27-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})

Theoremnbumgr 26288* The set of neighbors of an arbitrary class in a multigraph. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ UMGraph → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})

Theoremnbusgrvtx 26289* The set of neighbors of a vertex in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})

Theoremnbusgr 26290* The set of neighbors of an arbitrary class in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})

Theoremnbgr2vtx1edg 26291* If a graph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) (Revised by AV, 25-Mar-2021.) (Proof shortened by AV, 13-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((#‘𝑉) = 2 ∧ 𝑉𝐸) → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))

Theoremnbuhgr2vtx1edgblem 26292* Lemma for nbuhgr2vtx1edgb 26293. This reverse direction of nbgr2vtx1edg 26291 only holds for classes whose edges are subsets of the set of vertices, which is the property of hypergraphs. (Contributed by AV, 2-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)

Theoremnbuhgr2vtx1edgb 26293* If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 2) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))

Theoremnbusgreledg 26294 A class/vertex is a neighbor of another class/vertex in a simple graph iff the vertices are endpoints of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.) (Revised by AV, 26-Oct-2020.)
𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸))

Theoremuhgrnbgr0nb 26295* A vertex which is not endpoint of an edge has no neighbor in a hypergraph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)

Theoremnbgr0vtxlem 26296* Lemma for nbgr0vtx 26297 and nbgr0edg 26298. (Contributed by AV, 15-Nov-2020.)
(𝜑 → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)       (𝜑 → (𝐺 NeighbVtx 𝐾) = ∅)

Theoremnbgr0vtx 26297 In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Theoremnbgr0edg 26298 In an empty graph (with no edges), every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Theoremnbgr1vtx 26299 In a graph with one vertex, all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
((#‘(Vtx‘𝐺)) = 1 → (𝐺 NeighbVtx 𝐾) = ∅)

Theoremnbgrnself 26300* A vertex in a graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)

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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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