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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuslgra1 25901 The graph with one edge, analogous to umgra1 25855. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → 𝑉 USLGrph {⟨𝐴, {𝐵, 𝐶}⟩})

Theoremusgra1 25902 The graph with one edge, analogous to umgra1 25855 ( with additional assumption that 𝐵𝐶 since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → (𝐵𝐶𝑉 USGrph {⟨𝐴, {𝐵, 𝐶}⟩}))

Theoremuslgraun 25903 The union of two simple graphs with loops (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are (simple) graphs (with loops), then 𝑉, 𝐸𝐹 is a multigraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices), analogous to umgraun 25857. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑉 USLGrph 𝐸)    &   (𝜑𝑉 USLGrph 𝐹)       (𝜑𝑉 UMGrph (𝐸𝐹))

Theoremusgraedg2 25904 The value of the "edge function" of a graph is a set containing two elements (the vertices the corresponding edge is connecting), analogous to umgrale 25850. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸) → (#‘(𝐸𝑋)) = 2)

Theoremusgraedgprv 25905 In an undirected graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} → (𝑀𝑉𝑁𝑉)))

Theoremusgraedgrnv 25906 An edge of an undirected simple graph always connects two vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
((𝑉 USGrph 𝐸 ∧ {𝑀, 𝑁} ∈ ran 𝐸) → (𝑀𝑉𝑁𝑉))

Theoremusgranloopv 25907 In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
((𝑉 USGrph 𝐸𝑀𝑊) → ((𝐸𝑋) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremusgranloop 25908* In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.)
(𝑉 USGrph 𝐸 → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremusgranloop0 25909* A simple undirected graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉) → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) = {𝑈}} = ∅)

Theoremusgraedgrn 25910 An edge of an undirected simple graph without loops always connects two different vertices. (Contributed by Alexander van der Vekens, 2-Sep-2017.)
((𝑉 USGrph 𝐸 ∧ {𝑀, 𝑁} ∈ ran 𝐸) → 𝑀𝑁)

Theoremusgra2edg 25911* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
(((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ∃𝑥 ∈ dom 𝐸𝑦 ∈ dom 𝐸(𝑥𝑦𝑁 ∈ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑦)))

Theoremusgra2edg1 25912* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
(((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))

Theoremusgrarnedg 25913* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
((𝑉 USGrph 𝐸𝑌 ∈ ran 𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑌 = {𝑎, 𝑏}))

Theoremedgprvtx 25914* An edge of an undirected simple graph without loops is a proper unordered pair of vertices. (Contributed by AV, 1-Jan-2020.)
((𝐺 ∈ USGrph ∧ 𝐸 ∈ (Edges‘𝐺)) → ∃𝑥 ∈ (1st𝐺)∃𝑦 ∈ (1st𝐺)(𝑥𝑦𝐸 = {𝑥, 𝑦}))

Theoremusgraedg3 25915* The value of the "edge function" of a graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.)
((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸) → ∃𝑥𝑉𝑦𝑉 (𝑥𝑦 ∧ (𝐸𝑋) = {𝑥, 𝑦}))

Theoremusgraedg4 25916* The value of the "edge function" of a graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.)
((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})

Theoremusgraedgreu 25917* The value of the "edge function" of a graph is a uniquely determined set containing two elements (the endvertices of the corresponding edge). Concretising usgraedg4 25916. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃!𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})

Theoremusgrarnedg1 25918* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
((𝑉 USGrph 𝐸 ∧ ∃𝑦 ∈ ran 𝐸 𝑦 = (𝐸𝐼)) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏 ∧ (𝐸𝐼) = {𝑎, 𝑏}))

Theoremusgra1v 25919 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.)
({𝐴} USGrph 𝐸𝐸 = ∅)

Theoremusgraidx2vlem1 25920* Lemma 1 for usgraidx2v 25922. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}       ((𝑉 USGrph 𝐸𝑌𝐴) → (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) ∈ 𝑉)

Theoremusgraidx2vlem2 25921* Lemma 2 for usgraidx2v 25922. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}       ((𝑉 USGrph 𝐸𝑌𝐴) → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁}))

Theoremusgraidx2v 25922* The mapping of indices of edges containing a given vertex into the set of vertices is 1-1. The index is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}    &   𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}))       ((𝑉 USGrph 𝐸𝑁𝑉) → 𝐹:𝐴1-1𝑉)

Theoremusgraedgleord 25923* In a 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.)
((𝑉 USGrph 𝐸𝑁𝑉) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≤ (#‘𝑉))

17.1.3.2  Undirected simple graphs - examples

Theoremusgraex0elv 25924 Lemma 0 for usgraexmpl 25930. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
𝑉 = (0...4)       0 ∈ 𝑉

Theoremusgraex1elv 25925 Lemma 1 for usgraexmpl 25930. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
𝑉 = (0...4)       1 ∈ 𝑉

Theoremusgraex2elv 25926 Lemma 2 for usgraexmpl 25930. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
𝑉 = (0...4)       2 ∈ 𝑉

Theoremusgraex3elv 25927 Lemma 3 for usgraexmpl 25930. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
𝑉 = (0...4)       3 ∈ 𝑉

Theoremusgraexmpldifpr 25928 Lemma for usgraexmpl 25930: 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}))

Theoremusgraexmplef 25929* Lemma for usgraexmpl 25930. (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}

Theoremusgraexmpl 25930 𝑉, 𝐸 is a 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.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩       𝑉 USGrph 𝐸

Theoremusgraexmplvtx 25931 The vertices 0, 1, 2, 3, 4 of the graph 𝑉, 𝐸. (Contributed by AV, 12-Jan-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩       (𝑉1st 𝐸) = ({0, 1, 2} ∪ {3, 4})

Theoremusgraexmpledg 25932 The edges {0, 1}, {1, 2}, {2, 0}, {0, 3} of the graph 𝑉, 𝐸. (Contributed by AV, 12-Jan-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩       (𝑉Edges𝐸) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})

Theoremusgraexmplc 25933 𝐺 = ⟨𝑉, 𝐸 is a graph of five vertices 0, 1, 2, 3, 4, with edges {0, 1}, {1, 2}, {2, 0}, {0, 3}. (Contributed by AV, 12-Jan-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐺 ∈ USGrph

Theoremusgraexmplcvtx 25934 The vertices 0, 1, 2, 3, 4 of the graph 𝐺. (Contributed by AV, 12-Jan-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (1st𝐺) = ({0, 1, 2} ∪ {3, 4})

Theoremusgraexmplcedg 25935 The edges {0, 1}, {1, 2}, {2, 0}, {0, 3} of the graph 𝐺. (Contributed by AV, 12-Jan-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (Edges‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})

17.1.3.3  Finite undirected simple graphs

Theoremfiusgraedgfi 25936* In a finite graph the number of edges which contain a given vertex is also finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ Fin)

Theoremusgrafisindb0 25937 The size of a finite simple graph with 0 vertices is 0. Used for the base case of the induction in usgrafis 25944. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
((𝑉 USGrph 𝐸 ∧ (#‘𝑉) = 0) → (#‘𝐸) = 0)

Theoremusgrafisindb1 25938 The size of a finite simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
((𝑉 USGrph 𝐸 ∧ (#‘𝑉) = 1) → (#‘𝐸) = 0)

Theoremusgrares1 25939* Restricting an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       ((𝑉 USGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)

Theoremusgrafilem1 25940* The domain of the edge function is the union of the arguments/indices of all edges containing a specific vertex and the arguments/indices of all edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       dom 𝐸 = (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})

Theoremusgrafilem2 25941* In a graph with a finite number of vertices, the number of edges is finite if and only if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin))

Theoremusgrafisinds 25942* In a graph with a finite number of vertices, the number of edges is finite if the number of edges not containing one of the vertices is finite. Used for the step of the induction in usgrafis 25944. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       (𝑌 ∈ ℕ0 → ((𝑉 USGrph 𝐸 ∧ (#‘𝑉) = 𝑌𝑁𝑉) → (𝐹 ∈ Fin → 𝐸 ∈ Fin)))

Theoremusgrafisbase 25943 Induction base for usgrafis 25944. Main work is done in usgrafisindb0 25937. (Contributed by Alexander van der Vekens, 5-Jan-2018.)
((𝑉 USGrph 𝐸 ∧ (#‘𝑉) = 0) → 𝐸 ∈ Fin)

Theoremusgrafis 25944 A simple undirected graph with a finite number of vertices has also only a finite number of edges. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin) → 𝐸 ∈ Fin)

Theoremusgrafiedg 25945 A simple undirected graph with a finite number of vertices has also only a finite number of edges. (Contributed by AV, 2-Jan-2020.)
((𝐺 ∈ USGrph ∧ (1st𝐺) ∈ Fin) → (Edges‘𝐺) ∈ Fin)

17.1.4  Neighbors, complete graphs and universal vertices

Syntaxcnbgra 25946 Extend class notation with Neighbors (of a vertex in a graph).
class Neighbors

Syntaxccusgra 25947 Extend class notation with complete (undirected simple) graphs.
class ComplUSGrph

Syntaxcuvtx 25948 Extend class notation with the universal vertices (in a graph).
class UnivVertex

Definitiondf-nbgra 25949* Define the class of all Neighbors of a vertex in a graph. The neighbors of a vertex are all vertices which are connected with this vertex by an edge. (Contributed by Alexander van der Vekens and Mario Carneiro, 7-Oct-2017.)
Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st𝑔) ↦ {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)})

Definitiondf-cusgra 25950* Define the class of all complete (undirected simple) graphs. An undirected simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
ComplUSGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒)}

Definitiondf-uvtx 25951* Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})

17.1.4.1  Neighbors

Theoremnbgraop 25952* The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
(((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})

TheoremnbgraopALT 25953* Alternate proof of nbgraop 25952 using mpt2xopoveq 7232, but being longer. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})

Theoremnbgraop1 25954* The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
(((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (Fun 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸𝑖) = {𝑁, 𝑛}}))

Theoremnbgrael 25955 The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))

Theoremnbgranv0 25956 There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)

Theoremnbusgra 25957* The set of neighbors of a vertex in a graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Proof shortened by Alexander van der Vekens, 25-Jan-2018.)
(𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})

Theoremnbgra0nb 25958* A vertex which is not endpoint of an edge has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑉 USGrph 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁𝑥 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅))

Theoremnbgraeledg 25959 A class/vertex is a neighbor of another class/vertex if and only if it is an endpoint of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
(𝑉 USGrph 𝐸 → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ {𝑁, 𝐾} ∈ ran 𝐸))

Theoremnbgraisvtx 25960 Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑉 USGrph 𝐸 → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) → 𝑁𝑉))

Theoremnbgra0edg 25961 In a graph with no edges, every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑉 USGrph ∅ → (⟨𝑉, ∅⟩ Neighbors 𝐾) = ∅)

Theoremnbgrassvt 25962 The neighbors of a vertex in a graph are a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ 𝑉)

Theoremnbgranself 25963* A vertex in a graph (without loops!) is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑉 USGrph 𝐸 → ∀𝑣𝑉 𝑣 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑣))

Theoremnbgrassovt 25964 The neighbors of a vertex are a subset of the other vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑉 USGrph 𝐸 → (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁})))

Theoremnbgranself2 25965 A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑉 USGrph 𝐸𝑁 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))

Theoremnbgrassvwo 25966 The neighbors of a vertex in a graph are a subset of all vertices of the graph except the vertex itself. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
(𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Theoremnbgrassvwo2 25967 The neighbors of a vertex in a graph are a subset of all vertices of the graph except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
((𝑉 USGrph 𝐸𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))

Theoremnbgrasym 25968 A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑉 USGrph 𝐸 → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ 𝐾 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))

Theoremnbgracnvfv 25969 Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
((𝑉 USGrph 𝐸𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈)) → (𝐸‘(𝐸‘{𝑈, 𝑁})) = {𝑈, 𝑁})

Theoremnbgraf1olem1 25970* Lemma 1 for nbgraf1o 25976. For each neighbor of a vertex there is exactly one index for the edge between the vertex and its neighbor. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑀𝑁) → ∃!𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀})

Theoremnbgraf1olem2 25971* Lemma 2 for nbgraf1o 25976. The mapping of neighbors to edge indices is a function. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐹:𝑁𝐼)

Theoremnbgraf1olem3 25972* Lemma 3 for nbgraf1o 25976. The restricted iota of an edge is the function value of the converse applied to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀}) = (𝐸‘{𝑈, 𝑀}))

Theoremnbgraf1olem4 25973* Lemma 4 for nbgraf1o 25976. The mapping of neighbors to edge indices applied on a neighbor is the function value of the converse applied on the edge between the vertex and this neighbor. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝐹𝑀) = (𝐸‘{𝑈, 𝑀}))

Theoremnbgraf1olem5 25974* Lemma 5 for nbgraf1o 25976. The mapping of neighbors to edge indices is a one-to-one onto function. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}    &   𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))       ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐹:𝑁1-1-onto𝐼)

Theoremnbgraf1o0 25975* The set of neighbors of a vertex is isomorphic to the set of indices of edges containing the vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)    &   𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}       ((𝑉 USGrph 𝐸𝑈𝑉) → ∃𝑓 𝑓:𝑁1-1-onto𝐼)

Theoremnbgraf1o 25976* The set of neighbors of a vertex is isomorphic to the set of indices of edges containing the vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑈)–1-1-onto→{𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)})

Theoremnbusgrafi 25977 The class of neighbors of a vertex in a finite graph is a finite set. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
((𝑉 USGrph 𝐸𝑁𝑉𝐸 ∈ Fin) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∈ Fin)

Theoremnbfiusgrafi 25978 The class of neighbors of a vertex in a finite graph is a finite set. (Contributed by Alexander van der Vekens, 7-Mar-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∈ Fin)

Theoremedgusgranbfin 25979* The number of neighbors of a vertex in a graph is finite, if and only if the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑈) ∈ Fin ↔ {𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)} ∈ Fin))

Theoremnb3graprlem1 25980 Lemma 1 for nb3grapr 25982. (Contributed by Alexander van der Vekens, 15-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))

Theoremnb3graprlem2 25981* Lemma 2 for nb3grapr 25982. (Contributed by Alexander van der Vekens, 17-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣})(⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑣, 𝑤}))

Theoremnb3grapr 25982* The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))

Theoremnb3grapr2 25983 The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵})))

Theoremnb3gra2nb 25984 If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶}) ↔ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵})))

17.1.4.2  Complete graphs

Theoremiscusgra 25985* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))

Theoremiscusgra0 25986* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
(𝑉 ComplUSGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))

Theoremcusisusgra 25987 A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
(𝑉 ComplUSGrph 𝐸𝑉 USGrph 𝐸)

Theoremcusgrarn 25988* In a complete simple graph, the range of the edge function consists of all the pairs with different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
(𝑉 ComplUSGrph 𝐸 → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremcusgra0v 25989 A graph with no vertices (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
∅ ComplUSGrph ∅

Theoremcusgra1v 25990 A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
{𝐴} ComplUSGrph ∅

Theoremcusgra2v 25991 A graph with two (different) vertices is complete if and only if there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐵} ComplUSGrph 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸)))

Theoremnbcusgra 25992 In a complete (undirected simple) graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))

Theoremcusgra3v 25993 A graph with three (different) vertices is complete if and only if there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
𝑉 = {𝐴, 𝐵, 𝐶}       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝑉 ComplUSGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))

Theoremcusgra3vnbpr 25994* The neighbors of a vertex in a graph with three elements are unordered pairs of the other vertices if and only if the graph is complete. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
𝑉 = {𝐴, 𝐵, 𝐶}       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝑉 USGrph 𝐸 ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝑉 ComplUSGrph 𝐸 ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(⟨𝑉, 𝐸⟩ Neighbors 𝑥) = {𝑦, 𝑧}))

Theoremcusgraexilem1 25995* Lemma 1 for cusgraexi 25997. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)

Theoremcusgraexilem2 25996* Lemma 2 for cusgraexi 25997. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊𝑉 USGrph ( I ↾ 𝑃))

Theoremcusgraexi 25997* For each set the identity function restricted to the set of pairs of elements from the given set is an edge function, so that the given set together with this edge function is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊𝑉 ComplUSGrph ( I ↾ 𝑃))

Theoremcusgraexg 25998* For each set there is an edge function so that the set together with this edge function is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
(𝑉𝑊 → ∃𝑒 𝑉 ComplUSGrph 𝑒)

Theoremcusgrasizeindb0 25999 Base case of the induction in cusgrasize 26006. The size of a complete simple graph with 0 vertices is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
((𝑉 ComplUSGrph 𝐸 ∧ (#‘𝑉) = 0) → (#‘𝐸) = ((#‘𝑉)C2))

Theoremcusgrasizeindb1 26000 Base case of the induction in cusgrasize 26006. The size of a complete simple graph with 1 vertex is 0=((1-1)*1)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
((𝑉 ComplUSGrph 𝐸 ∧ (#‘𝑉) = 1) → (#‘𝐸) = ((#‘𝑉)C2))

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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 42301-42360
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