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Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremusgredg2vlem1 27601* Lemma 1 for usgredg2v 27603. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}       ((𝐺 ∈ USGraph ∧ 𝑌𝐴) → (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
 
Theoremusgredg2vlem2 27602* Lemma 2 for usgredg2v 27603. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}       ((𝐺 ∈ USGraph ∧ 𝑌𝐴) → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁}))
 
Theoremusgredg2v 27603* 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 𝐸𝑁 ∈ (𝐸𝑥)}    &   𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}))       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
 
Theoremusgriedgleord 27604* Alternate version of usgredgleord 27609, 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 ∧ 𝑁𝑉) → (♯‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≤ (♯‘𝑉))
 
Theoremushgredgedg 27605* 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 𝐼𝑁 ∈ (𝐼𝑖)}    &   𝐵 = {𝑒𝐸𝑁𝑒}    &   𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))       ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
 
Theoremusgredgedg 27606* 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 𝐼𝑁 ∈ (𝐼𝑖)}    &   𝐵 = {𝑒𝐸𝑁𝑒}    &   𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
 
Theoremushgredgedgloop 27607* 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 ∣ (𝐼𝑖) = {𝑁}}    &   𝐵 = {𝑒𝐸𝑒 = {𝑁}}    &   𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))       ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
 
Theoremuspgredgleord 27608* 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 ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) ≤ (♯‘𝑉))
 
Theoremusgredgleord 27609* 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 ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) ≤ (♯‘𝑉))
 
TheoremusgredgleordALT 27610* Alternate proof for usgredgleord 27609 based on usgriedgleord 27604. 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 ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) ≤ (♯‘𝑉))
 
Theoremusgrstrrepe 27611* Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (𝜑𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑𝐺 ∈ V). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.)
𝑉 = (Base‘𝐺)    &   𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct 𝑋)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐸𝑊)    &   (𝜑𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})       (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph)
 
16.2.6  Examples for graphs
 
Theoremusgr0e 27612 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.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ USGraph)
 
Theoremusgr0vb 27613 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.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))
 
Theoremuhgr0v0e 27614 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 = ∅)
 
Theoremuhgr0vsize0 27615 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)
 
Theoremuhgr0edgfi 27616 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)
 
Theoremusgr0v 27617 The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph)
 
Theoremuhgr0vusgr 27618 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)
 
Theoremusgr0 27619 The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
∅ ∈ USGraph
 
Theoremuspgr1e 27620 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)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})       (𝜑𝐺 ∈ USPGraph)
 
Theoremusgr1e 27621 A simple graph with one edge (with additional assumption that 𝐵𝐶 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)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})    &   (𝜑𝐵𝐶)       (𝜑𝐺 ∈ USGraph)
 
Theoremusgr0eop 27622 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.)
(𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)
 
Theoremuspgr1eop 27623 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.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ USPGraph)
 
Theoremuspgr1ewop 27624 A simple pseudograph with (at least) two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
((𝑉𝑊𝐴𝑉𝐵𝑉) → ⟨𝑉, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USPGraph)
 
Theoremuspgr1v1eop 27625 A simple pseudograph with (at least) one vertex and one edge (a loop). (Contributed by AV, 5-Dec-2020.)
((𝑉𝑊𝐴𝑋𝐵𝑉) → ⟨𝑉, {⟨𝐴, {𝐵}⟩}⟩ ∈ USPGraph)
 
Theoremusgr1eop 27626 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.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → (𝐵𝐶 → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ USGraph))
 
Theoremuspgr2v1e2w 27627 A simple pseudograph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
((𝐴𝑋𝐵𝑌) → ⟨{𝐴, 𝐵}, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USPGraph)
 
Theoremusgr2v1e2w 27628 A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
((𝐴𝑋𝐵𝑌𝐴𝐵) → ⟨{𝐴, 𝐵}, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USGraph)
 
Theoremedg0usgr 27629 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.)
((𝐺𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)
 
Theoremlfuhgr1v0e 27630* 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 𝐼𝐸) → (Edg‘𝐺) = ∅)
 
Theoremusgr1vr 27631 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.)
((𝐴𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅))
 
Theoremusgr1v 27632 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.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))
 
Theoremusgr1v0edg 27633 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.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = {𝐴} ∧ Fun (iEdg‘𝐺)) → (𝐺 ∈ USGraph ↔ (Edg‘𝐺) = ∅))
 
Theoremusgrexmpldifpr 27634 Lemma for usgrexmpledg 27638: 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}))
 
Theoremusgrexmplef 27635* Lemma for usgrexmpl 27639. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩       𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
 
Theoremusgrexmpllem 27636 Lemma for usgrexmpl 27639. (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)
 
Theoremusgrexmplvtx 27637 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})
 
Theoremusgrexmpledg 27638 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}})
 
Theoremusgrexmpl 27639 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.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐺 ∈ USGraph
 
Theoremgriedg0prc 27640* 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 27641* 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 27642 The class of simple graphs is a proper class (and therefore, because of prcssprc 5250, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.)
USGraph ∉ V
 
16.2.7  Subgraphs
 
Syntaxcsubgr 27643 Extend class notation with subgraphs.
class SubGraph
 
Definitiondf-subgr 27644* 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 27645 The class of the subgraph relation is a relation. (Contributed by AV, 16-Nov-2020.)
Rel SubGraph
 
Theoremsubgrv 27646 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 27647 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 27648 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 27649 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 27650 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 27651 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 27652 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 27653 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 27654 The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
(𝐺𝑊 → ∅ SubGraph 𝐺)
 
Theorem0uhgrsubgr 27655 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 27656 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 27657 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 27658 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 27659 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 27660 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 27661* 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 27662 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 27663 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 27664 A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.)
((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UMGraph)
 
Theoremsubusgr 27665 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 27666 Lemma for uhgrspansubgr 27667: 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 27667. (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 27667 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 27668 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 27669 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 27670 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 27671 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 27672 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 27673 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 27674 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 27675 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 27676 Lemma 1 for uhgrspan1 27679. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}       ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
 
Theoremuhgrspan1lem2 27677 Lemma 2 for uhgrspan1 27679. (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 27678 Lemma 3 for uhgrspan1 27679. (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 27679* 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 27680* Lemma for upgrres 27682. (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 27681* Lemma for umgrres 27683 and usgrres 27684. (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 27682* A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 27679) 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 27683* A subgraph obtained by removing one vertex and all edges incident with this vertex from a multigraph (see uhgrspan1 27679) 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 27684* A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 27679) 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 27685* Lemma 1 for upgrres1 27689. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)
 
Theoremumgrres1lem 27686* Lemma for umgrres1 27690. (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 27687* Lemma 2 for upgrres1 27689. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
 
Theoremupgrres1lem3 27688* Lemma 3 for upgrres1 27689. (Contributed by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (iEdg‘𝑆) = ( I ↾ 𝐹)
 
Theoremupgrres1 27689* 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 27644 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 27690* 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 27644 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 27691* 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 27644 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 27692 Extend class notation with finite simple graphs.
class FinUSGraph
 
Definitiondf-fusgr 27693 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 27694 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 27695 A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
 
Theoremisfusgrf1 27696* 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 27697 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 27698 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 27699 A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.)
((𝑉𝑋𝐸𝑌) → (⟨𝑉, 𝐸⟩ ∈ FinUSGraph ↔ (⟨𝑉, 𝐸⟩ ∈ USGraph ∧ 𝑉 ∈ Fin)))
 
Theoremusgredgffibi 27700 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))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46532
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