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Theorem List for Metamath Proof Explorer - 40301-40400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxnn0xr 40301 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ∈ ℝ*)

Theoremxnn0xrge0 40302 An extended nonnegative integer is an extended nonnegative real. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ∈ (0[,]+∞))

Theorem0xnn0 40303 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
0 ∈ ℕ0*

Theorempnf0xnn0 40304 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
+∞ ∈ ℕ0*

Theoremnn0nepnf 40305 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0𝐴 ≠ +∞)

Theoremnn0xnn0d 40306 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℕ0*)

Theoremnn0nepnfd 40307 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ≠ +∞)

Theoremxnn0nemnf 40308 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ≠ -∞)

Theoremxnn0xrnemnf 40309 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ*𝐴 ≠ -∞))

Theoremxnn0ge0 40310 An extended nonnegative integer is greater than or equal to 0, see also nn0pnfge0 11716. (Contributed by AV, 10-Dec-2020.) (Proof modification is discouraged.)
(𝑁 ∈ ℕ0* → 0 ≤ 𝑁)

Theoremxnn0nnn0pnf 40311 An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)

Theoremxnn0xaddcl 40312 The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → (𝐴 +𝑒 𝐵) ∈ ℕ0*)

Theoremxnn0add4d 40313 Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 11875. (Contributed by AV, 12-Dec-2020.)
(𝜑𝐴 ∈ ℕ0*)    &   (𝜑𝐵 ∈ ℕ0*)    &   (𝜑𝐶 ∈ ℕ0*)    &   (𝜑𝐷 ∈ ℕ0*)       (𝜑 → ((𝐴 +𝑒 𝐵) +𝑒 (𝐶 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐶) +𝑒 (𝐵 +𝑒 𝐷)))

Theoremxnn0xadd0 40314 The sum of two extended nonnegative integers is 0 iff each of the two extended nonnegative integers is 0. (Contributed by AV, 14-Dec-2020.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))

Theoremxnn0n0n1ge2b 40315 An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by AV, 5-Apr-2021.)
(𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁))

Theoremhashfxnn0 40316 The size function is a function into the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
#:V⟶ℕ0*

Theoremhashxnn0 40317 The value of the hash function for a set is an extended nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Dec-2020.)
(𝑀𝑉 → (#‘𝑀) ∈ ℕ0*)

20.34.7.19  Finite and infinite sums - extension

Theoremfsummsndifre 40318* A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ)

Theoremfsumsplitsndif 40319* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ 𝑋𝐴 ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘𝐴 𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 + 𝑋 / 𝑘𝐵))

Theoremfsummmodsndifre 40320* A finite sum of summands modulo a positive number with one of its summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})(𝐵 mod 𝑁) ∈ ℝ)

Theoremfsummmodsnunz 40321* A finite sum of summands modulo a positive number with an additional summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) ∈ ℤ)

20.34.8  Graph theory (revised)

20.34.8.1  The edge function extractor for extensible structures

Syntaxcedgf 40322 Extend class notation with an edge function.
class .ef

Definitiondf-edgf 40323 Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)
.ef = Slot 18

Theoremedgfndxnn 40324 The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.)
(.ef‘ndx) ∈ ℕ

Theoremedgfndxid 40325 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.)
(𝐺𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))

Theorembaseltedgf 40326 The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.)
(Base‘ndx) < (.ef‘ndx)

Theoremslotsbaseefdif 40327 The slots Base and .ef are different. (Contributed by AV, 21-Sep-2020.)
(Base‘ndx) ≠ (.ef‘ndx)

20.34.8.2  Vertices and edges

The key concepts in graph theory are vertices and edges. In general, a graph "consists" (at least) of two sets: the set of vertices and the set of edges. The edges "connect" vertices. The meaning of "connect" is different for different kinds of graphs (directed/undirected graphs, hyper-/multi-/simple graphs, etc.). The simplest way to represent a graph (of any kind) is to define a graph as "an ordered pair of disjoint sets (V, E)" (see section I.1 in [Bollobas] p. 1), or in the notation of Metamath: 𝑉, 𝐸. Another way is to regard a graph as a mathematical structure, which can be enhanced by additional features (see Wikipedia "Mathematical structure", 24-Sep-2020, https://en.wikipedia.org/wiki/Mathematical_structure): "In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.". Such structures are provided as "extensible structures" in Metamath, see df-struct 15584.

To allow for expressing and proving most of the theorems for graphs independently from their representation, the functions Vtx and iEdg are defined (see df-vtx 40330 and df-iedg 40331), which provide the vertices resp. (indexed) edges of an arbitrary class 𝐺 which represents a graph: (Vtx‘𝐺) resp. (iEdg‘𝐺). Instead of providing edges themselves, iEdg is intended to provide a function as mapping of "indices" (the domain of the function) to the edges (therefore called "set of indexed edges"), which allows for hyper-/pseudo-/multigraphs with more than one edge between two (or more) vertices. In literature, these functions are often denoted also by "V" and "E", see section I.1 in [Bollobas] p. 1 ("If G is a graph, then V = V(G) is the vertex set of G, and E = E(G) is the edge set.") or section 1.1 in [Diestel] p. 2 ("The vertex set of graph G is referred to as V(G), its edge set as E(G)."). For example, e1 = e(1) = { a, b } and e2 = e(2) = { a, b } are two different edges connecting the same two vertices a and b (in a pseudograph). In section 1.10 of [Diestel] p. 28, the edge function is defined differently: as "map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.". Here, the domain is the set of abstract edges: for two different edges e1 and e2 connecting the same two vertices a and b, we would have e(e1) = e(e2) = { a, b }. Since the set of abstract edges can be chosen as index set, these definitions are equivalent.

The result of these functions are as expected: for a graph represented as ordered pair (𝐺 ∈ (V × V)), the set of vertices is (Vtx‘𝐺) = (1st𝐺) (see opvtxval 40335) and the set of (indexed) edges is (iEdg‘𝐺) = (2nd𝐺) (see opiedgval 40338), or if 𝐺 is given as ordered pair 𝐺 = ⟨𝑉, 𝐸, the set of vertices is (Vtx‘𝐺) = 𝑉 (see opvtxfv 40336) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see opiedgfv 40339).

And for a graph represented as extensible structure (𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩), the set of vertices is (Vtx‘𝐺) = (Base‘𝐺) (see funvtxval 40350) and the set of (indexed) edges is (iEdg‘𝐺) = (.ef‘𝐺) (see funiedgval 40351), or if 𝐺 is given in its simplest form as extensible structure with two slots (𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), the set of vertices is (Vtx‘𝐺) = 𝑉 (see struct2grvtx 40359) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see struct2griedg 40360).

These two representations are convertible, see graop 40361 and grastruct 40362: If 𝐺 is a graph (for example 𝐺 = ⟨𝑉, 𝐸), then 𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩} represents essentially the same graph, and if 𝐺 is a graph (for example 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), then 𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ represents essentially the same graph. In both cases, (Vtx‘𝐺) = (Vtx‘𝐻) and (iEdg‘𝐺) = (iEdg‘𝐻) hold. Theorems gropd 40363 and gropeld 40365 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. Analogously, theorems grstructd 40364 and grstructeld 40366 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then any extensible structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is also such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property.

Besides the usual way to represent graphs without edges (consisting of unconnected vertices only), which would be 𝐺 = ⟨𝑉, ∅⟩ or 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), ∅⟩}, a structure without a slot for edges can be used: 𝐺 = {⟨(Base‘ndx), 𝑉⟩}, see snstrvtxval 40367 and snstriedgval 40368. Analogously, the empty set can be used to represent the null graph, see vtxval0 40369 and iedgval0 40370, which can also be represented by 𝐺 = ⟨∅, ∅⟩ or 𝐺 = {⟨(Base‘ndx), ∅⟩, ⟨(.ef‘ndx), ∅⟩}. Even proper classes can be used to represent the null graph, see vtxvalprc 40375 and iedgvalprc 40376.

Other classes should not be used to represent graphs, because there could be a degenerated behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 40371 resp. iedgvalsnop 40372, and vtxval3sn 40373 resp. iedgval3sn 40374.

Syntaxcvtx 40328 Extend class notation with the vertices of "graphs".
class Vtx

Syntaxciedg 40329 Extend class notation with the indexed edges of "graphs".
class iEdg

Definitiondf-vtx 40330 Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)
Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))

Definitiondf-iedg 40331 Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)
iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)))

Theoremvtxval 40332 The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
(𝐺𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))

Theoremiedgval 40333 The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)
(𝐺𝑉 → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))

Theorem1vgrex 40334 A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉𝐺 ∈ V)

Theoremopvtxval 40335 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
(𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st𝐺))

Theoremopvtxfv 40336 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Theoremopvtxov 40337 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (𝑉Vtx𝐸) = 𝑉)

Theoremopiedgval 40338 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)
(𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd𝐺))

Theoremopiedgfv 40339 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Theoremopiedgov 40340 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (𝑉iEdg𝐸) = 𝐸)

Theoremopvtxfvi 40341 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
𝑉 ∈ V    &   𝐸 ∈ V       (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Theoremopiedgfvi 40342 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
𝑉 ∈ V    &   𝐸 ∈ V       (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸

Theoremfunvtxdm2val 40343 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgdm2val 40344 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theoremfunvtxval0 40345 The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.)
𝑆 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfunvtxdmge2val 40346 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgdmge2val 40347 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorembasvtxval 40348 The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → 2 ≤ (#‘dom 𝐺))    &   (𝜑𝑉𝑌)    &   (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)

Theoremedgfiedgval 40349 The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → 2 ≤ (#‘dom 𝐺))    &   (𝜑𝐸𝑌)    &   (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)

Theoremfunvtxval 40350 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgval 40351 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theoremstructvtxvallem 40352 Lemma for structvtxval 40353 and structiedg0val 40354. (Contributed by AV, 23-Sep-2020.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (𝐺 ∈ V ∧ Fun 𝐺 ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺))

Theoremstructvtxval 40353 The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (Vtx‘𝐺) = 𝑉)

Theoremstructiedg0val 40354 The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅)

(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → 2 ≤ (#‘dom 𝐺))

Theoremstructgrssvtx 40356 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)

Theoremstructgrssiedg 40357 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 14-Oct-2020.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)

Theoremstruct2grstr 40358 A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩

Theoremstruct2grvtx 40359 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (Vtx‘𝐺) = 𝑉)

Theoremstruct2griedg 40360 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 23-Sep-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (iEdg‘𝐺) = 𝐸)

Theoremgraop 40361 Any representation of a graph 𝐺 (especially as extensible structure 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.)
𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩       ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Theoremgrastruct 40362 Any representation of a graph 𝐺 (especially as ordered pair 𝐺 = ⟨𝑉, 𝐸) is convertible in a representation of the graph as extensible structure. (Contributed by AV, 8-Oct-2020.)
𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩}       ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Theoremgropd 40363* If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 11-Oct-2020.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)       (𝜑[𝑉, 𝐸⟩ / 𝑔]𝜓)

Theoremgrstructd 40364* If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑𝑆𝑋)    &   (𝜑 → Fun (𝑆 ∖ {∅}))    &   (𝜑 → 2 ≤ (#‘dom 𝑆))    &   (𝜑 → (Base‘𝑆) = 𝑉)    &   (𝜑 → (.ef‘𝑆) = 𝐸)       (𝜑[𝑆 / 𝑔]𝜓)

Theoremgropeld 40365* If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 11-Oct-2020.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)

Theoremgrstructeld 40366* If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑𝑆𝑋)    &   (𝜑 → Fun (𝑆 ∖ {∅}))    &   (𝜑 → 2 ≤ (#‘dom 𝑆))    &   (𝜑 → (Base‘𝑆) = 𝑉)    &   (𝜑 → (.ef‘𝑆) = 𝐸)       (𝜑𝑆𝐶)

Theoremsnstrvtxval 40367 The set of vertices of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See vtxvalsnop 40371 for the (degenerated) case where 𝑉 = (Base‘ndx). (Contributed by AV, 23-Sep-2020.)
𝑉 ∈ V    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩}       (𝑉 ≠ (Base‘ndx) → (Vtx‘𝐺) = 𝑉)

Theoremsnstriedgval 40368 The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 40372 for the (degenerated) case where 𝑉 = (Base‘ndx). (Contributed by AV, 24-Sep-2020.)
𝑉 ∈ V    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩}       (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅)

Theoremvtxval0 40369 Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
(Vtx‘∅) = ∅

Theoremiedgval0 40370 Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
(iEdg‘∅) = ∅

Theoremvtxvalsnop 40371 Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.)
𝐵 ∈ V    &   𝐺 = {⟨𝐵, 𝐵⟩}       (Vtx‘𝐺) = {𝐵}

Theoremiedgvalsnop 40372 Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.)
𝐵 ∈ V    &   𝐺 = {⟨𝐵, 𝐵⟩}       (iEdg‘𝐺) = {𝐵}

Theoremvtxval3sn 40373 Degenerated case 3 for vertices: The set of vertices of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V       (Vtx‘{{{𝐴}}}) = {𝐴}

Theoremiedgval3sn 40374 Degenerated case 3 for edges: The set of indexed edges of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V       (iEdg‘{{{𝐴}}}) = {𝐴}

Theoremvtxvalprc 40375 Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
(𝐶 ∉ V → (Vtx‘𝐶) = ∅)

Theoremiedgvalprc 40376 Degenerated case 4 for edges: The set of indexed edges of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
(𝐶 ∉ V → (iEdg‘𝐶) = ∅)

20.34.8.3  Undirected hypergraphs

Syntaxcuhgr 40377 Extend class notation with undirected hypergraphs.
class UHGraph

Syntaxcushgr 40378 Extend class notation with undirected simple hypergraphs.
class USHGraph

Definitiondf-uhgr 40379* Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 8-Oct-2020.)
UHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}

Definitiondf-ushgr 40380* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function 𝑒 is an injective (one-to-one) function into subsets of the set of vertices 𝑣, representing the (one or more) vertices incident to the edge. This definition corresponds to definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subsets of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are non-empty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by AV, 8-Oct-2020.)
USHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}

Theoremisuhgr 40381 The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Theoremisushgr 40382 The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))

Theoremuhgrf 40383 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))

Theoremushgrf 40384 The edge function of an undirected simple hypergraph is a function into the power set of the set of vertices. (Contributed by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USHGraph → 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}))

Theoremuhgrss 40385 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)

Theoremuhgreq12g 40386 If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐹 = (iEdg‘𝐻)       (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph ))

Theoremuhgrfun 40387 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 15-Dec-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → Fun 𝐸)

Theoremuhgrn0 40388 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)

Theoremlpvtx 40389 The endpoints of a loop (which is an edge at index 𝐽) are two (identical) vertices 𝐴. (Contributed by AV, 1-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))

Theoremushgruhgr 40390 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
(𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph )

Theoremuhgruhgra 40391 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝑉 UHGrph 𝐸)

Theoremuhgrauhgr 40392 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
((𝑉 UHGrph 𝐸𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝐺𝑊𝐺 ∈ UHGraph ))

Theoremuhgrauhgrbi 40393 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
((𝐺𝑊𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝑉 UHGrph 𝐸𝐺 ∈ UHGraph ))

Theoremisuhgrop 40394 The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 9-Oct-2020.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Theoremuhgr0e 40395 The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ UHGraph )

Theoremuhgr0vb 40396 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 9-Oct-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))

Theoremuhgr0 40397 The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.)
∅ ∈ UHGraph

Theoremuhgrun 40398 The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.)
(𝜑𝐺 ∈ UHGraph )    &   (𝜑𝐻 ∈ UHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UHGraph )

Theoremuhgrunop 40399 The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If 𝑉, 𝐸 and 𝑉, 𝐹 are hypergraphs, then 𝑉, 𝐸𝐹 is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.)
(𝜑𝐺 ∈ UHGraph )    &   (𝜑𝐻 ∈ UHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph )

Theoremushgrun 40400 The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
(𝜑𝐺 ∈ USHGraph )    &   (𝜑𝐻 ∈ USHGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UHGraph )

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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 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