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Theorem List for Metamath Proof Explorer - 26501-26600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisuspgr 26501* The property of being a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))

Theoremisusgr 26502* The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))

Theoremuspgrf 26503* The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Theoremusgrf 26504* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})

Theoremisusgrs 26505* The property of being a simple graph, simplified version of isusgr 26502. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.) (Proof shortened by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))

Theoremusgrfs 26506* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 26504. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})

Theoremusgrfun 26507 The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.) (Revised by AV, 13-Oct-2020.)
(𝐺 ∈ USGraph → Fun (iEdg‘𝐺))

Theoremusgredgss 26508* The set of edges of a simple graph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
(𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})

Theoremedgusgr 26509 An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝐸) = 2))

Theoremisuspgrop 26510* The property of being an undirected simple pseudograph 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, 25-Nov-2021.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))

Theoremisusgrop 26511* The property of being an undirected simple graph 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, 30-Nov-2020.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}))

Theoremusgrop 26512 A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.) (Proof shortened by AV, 30-Nov-2020.)
(𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph)

Theoremisausgr 26513* The property of an unordered pair to be an alternatively defined simple graph, defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}       ((𝑉𝑊𝐸𝑋) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))

Theoremausgrusgrb 26514* The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}       ((𝑉𝑋𝐸𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph))

Theoremusgrausgri 26515* A simple graph represented by an alternatively defined simple graph. (Contributed by AV, 15-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}       (𝐻 ∈ USGraph → (Vtx‘𝐻)𝐺(Edg‘𝐻))

Theoremausgrumgri 26516* If an alternatively defined simple graph has the vertices and edges of an arbitrary graph, the arbitrary graph is an undirected multigraph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 25-Nov-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}       ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → 𝐻 ∈ UMGraph)

Theoremausgrusgri 26517* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 15-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}    &   𝑂 = {𝑓𝑓:dom 𝑓1-1→ran 𝑓}       ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph)

Theoremusgrausgrb 26518* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}    &   𝑂 = {𝑓𝑓:dom 𝑓1-1→ran 𝑓}       ((𝐻𝑊 ∧ (iEdg‘𝐻) ∈ 𝑂) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ 𝐻 ∈ USGraph))

Theoremusgredgop 26519 An edge of a simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.) (Revised by AV, 15-Oct-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 = (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} ↔ ⟨𝑋, {𝑀, 𝑁}⟩ ∈ 𝐸))

Theoremusgrf1o 26520 The edge function of a simple graph is a bijective function onto its range. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1-onto→ran 𝐸)

Theoremusgrf1 26521 The edge function of a simple graph is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→ran 𝐸)

Theoremuspgrf1oedg 26522 The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))

Theoremusgrss 26523 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸𝑋) ⊆ 𝑉)

Theoremuspgrushgr 26524 A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Theoremuspgrupgr 26525 A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)

Theoremuspgrupgrushgr 26526 A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.)
(𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))

Theoremusgruspgr 26527 A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)

Theoremusgrumgr 26528 A simple graph is an undirected multigraph. (Contributed by AV, 25-Nov-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)

Theoremusgrumgruspgr 26529 A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020.)
(𝐺 ∈ USGraph ↔ (𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph))

Theoremusgruspgrb 26530* A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)
(𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))

Theoremusgrupgr 26531 A simple graph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 20-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)

Theoremusgruhgr 26532 A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)

Theoremusgrislfuspgr 26533* A simple graph is a loop-free simple pseudograph. (Contributed by AV, 27-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))

Theoremuspgrun 26534 The union 𝑈 of two simple pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)
(𝜑𝐺 ∈ USPGraph)    &   (𝜑𝐻 ∈ USPGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UPGraph)

Theoremuspgrunop 26535 The union of two simple pseudographs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are simple pseudographs, then 𝑉, 𝐸𝐹 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ USPGraph)    &   (𝜑𝐻 ∈ USPGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph)

Theoremusgrun 26536 The union 𝑈 of two simple graphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a multigraph (not necessarily a simple graph!) with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
(𝜑𝐺 ∈ USGraph)    &   (𝜑𝐻 ∈ USGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UMGraph)

Theoremusgrunop 26537 The union of two simple graphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are simple graphs, then 𝑉, 𝐸𝐹 is a multigraph (not necessarily a simple graph!) - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.)
(𝜑𝐺 ∈ USGraph)    &   (𝜑𝐻 ∈ USGraph)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UMGraph)

Theoremusgredg2 26538 The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸𝑋)) = 2)

Theoremusgredg2ALT 26539 Alternate proof of usgredg2 26538, not using umgredg2 26448. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸𝑋)) = 2)

Theoremusgredgprv 26540 In a simple graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} → (𝑀𝑉𝑁𝑉)))

TheoremusgredgprvALT 26541 Alternate proof of usgredgprv 26540, using usgredg2 26538 instead of umgredgprv 26455. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} → (𝑀𝑉𝑁𝑉)))

Theoremusgredgppr 26542 An edge of a simple graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge). Analogue of usgredg2 26538. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → (♯‘𝐶) = 2)

Theoremusgrpredgv 26543 An edge of a simple graph always connects two vertices. Analogue of usgredgprv 26540. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀𝑉𝑁𝑉))

Theoremedgssv2 26544 An edge of a simple graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → (𝐶𝑉 ∧ (♯‘𝐶) = 2))

Theoremusgredg 26545* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Shortened by AV, 25-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝐶 = {𝑎, 𝑏}))

Theoremusgrnloopv 26546 In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀𝑊) → ((𝐸𝑋) = {𝑀, 𝑁} → 𝑀𝑁))

TheoremusgrnloopvALT 26547 Alternate proof of usgrnloopv 26546, not using umgrnloopv 26454. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀𝑊) → ((𝐸𝑋) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremusgrnloop 26548* In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑀, 𝑁} → 𝑀𝑁))

TheoremusgrnloopALT 26549* Alternate proof of usgrnloop 26548, not using umgrnloop 26456. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremusgrnloop0 26550* A simple graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) = {𝑈}} = ∅)

Theoremusgrnloop0ALT 26551* Alternate proof of usgrnloop0 26550, not using umgrnloop0 26457. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) = {𝑈}} = ∅)

Theoremusgredgne 26552 An edge of a simple graph always connects two different vertices. Analogue of usgrnloopv 26546 resp. usgrnloop 26548. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → 𝑀𝑁)

Theoremusgrf1oedg 26553 The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by by AV, 18-Oct-2020.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1-onto𝐸)

Theoremuhgr2edg 26554* If a vertex is adjacent to two different vertices in a hypergraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremumgr2edg 26555* If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremusgr2edg 26556* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremumgr2edg1 26557* If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 8-Jun-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼𝑥))

Theoremusgr2edg1 26558* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 8-Jun-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼𝑥))

Theoremumgrvad2edg 26559* If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex, analogous to usgr2edg 26556. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)
𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥𝐸𝑦𝐸 (𝑥𝑦𝑁𝑥𝑁𝑦))

Theoremumgr2edgneu 26560* If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex, analogous to usgr2edg1 26558. Lemma for theorems about friendship graphs. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥𝐸 𝑁𝑥)

Theoremusgrsizedg 26561 In a simple graph, the size of the edge function is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 7-Jun-2021.)
(𝐺 ∈ USGraph → (♯‘(iEdg‘𝐺)) = (♯‘(Edg‘𝐺)))

Theoremusgredg3 26562* The value of the "edge function" of a simple graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥𝑉𝑦𝑉 (𝑥𝑦 ∧ (𝐸𝑋) = {𝑥, 𝑦}))

Theoremusgredg4 26563* For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})

Theoremusgredgreu 26564* For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃!𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})

Theoremusgredg2vtx 26565* For a vertex incident to an edge there is another vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 5-Dec-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌𝐸) → ∃𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theoremuspgredg2vtxeu 26566* For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 6-Dec-2020.)
((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theoremusgredg2vtxeu 26567* For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theoremusgredg2vtxeuALT 26568* Alternate proof of usgredg2vtxeu 26567, using edgiedgb 26402, the general translation from (iEdg‘𝐺) to (Edg‘𝐺). (Contributed by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theoremuspgredg2vlem 26569* Lemma for uspgredg2v 26570. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐴 = {𝑒𝐸𝑁𝑒}       ((𝐺 ∈ USPGraph ∧ 𝑌𝐴) → (𝑧𝑉 𝑌 = {𝑁, 𝑧}) ∈ 𝑉)

Theoremuspgredg2v 26570* In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐴 = {𝑒𝐸𝑁𝑒}    &   𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 𝑦 = {𝑁, 𝑧}))       ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)

Theoremusgredg2vlem1 26571* Lemma 1 for usgredg2v 26573. (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 26572* Lemma 2 for usgredg2v 26573. (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 26573* 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 26574* Alternate version of usgredgleord 26580, 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 26575* 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 26576* 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 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 26577* 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)

TheoremushgredgedgloopOLD 26578* Obsolete version of ushgredgedgloop 26577 as of 6-Jul-2022. (Contributed by AV, 11-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)    &   𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}    &   𝐵 = {𝑒𝐸𝑒 = {𝑁}}    &   𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))       ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)

Theoremuspgredgleord 26579* 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 26580* 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 26581* Alternate proof for usgredgleord 26580 based on usgriedgleord 26574. 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 26582* 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 26583 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 26584 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 26585 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 26586 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 26587 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 26588 The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph)

Theoremuhgr0vusgr 26589 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 26590 The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
∅ ∈ USGraph

Theoremuspgr1e 26591 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 26592 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 26593 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 26594 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 26595 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 26596 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 26597 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 26598 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 26599 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 26600 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)

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