P. 292 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-cusgr 29101	Define the class of all complete simple graphs. A simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge, see definition in section 1.1 of [Diestel] p. 3. In contrast, the definition in section I.1 of [Bollobas] p. 3 is based on the size of (finite) complete graphs, see cusgrsize 29143. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by BJ, 14-Feb-2022.)
⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
Theorem	cplgruvtxb 29102	A graph 𝐺 is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 15-Feb-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
Theorem	prcliscplgr 29103*	A proper class (representing a null graph, see vtxvalprc 28737) has the property of a complete graph (see also cplgr0v 29116), but cannot be an element of ComplGraph, of course. Because of this, a sethood antecedent like 𝐺 ∈ 𝑊 is necessary in the following theorems like iscplgr 29104. (Contributed by AV, 14-Feb-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (¬ 𝐺 ∈ V → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
Theorem	iscplgr 29104*	The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
Theorem	iscplgrnb 29105*	A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Theorem	iscplgredg 29106*	A graph 𝐺 is complete iff all vertices are connected with each other by (at least) one edge. (Contributed by AV, 10-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})∃𝑒 ∈ 𝐸 {𝑣, 𝑛} ⊆ 𝑒))
Theorem	iscusgr 29107	The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
Theorem	cusgrusgr 29108	A complete simple graph is a simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
Theorem	cusgrcplgr 29109	A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.)
⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
Theorem	iscusgrvtx 29110*	A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
Theorem	cusgruvtxb 29111	A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉))
Theorem	iscusgredg 29112*	A simple graph is complete iff all vertices are connected by an edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 1-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ 𝐸))
Theorem	cusgredg 29113*	In a complete simple graph, the edges are all the pairs of different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 1-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ ComplUSGraph → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
Theorem	cplgr0 29114	The null graph (with no vertices and no edges) represented by the empty set is a complete graph. (Contributed by AV, 1-Nov-2020.)
⊢ ∅ ∈ ComplGraph
Theorem	cusgr0 29115	The null graph (with no vertices and no edges) represented by the empty set is a complete simple graph. (Contributed by AV, 1-Nov-2020.)
⊢ ∅ ∈ ComplUSGraph
Theorem	cplgr0v 29116	A null graph (with no vertices) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ ComplGraph)
Theorem	cusgr0v 29117	A graph with no vertices and no edges is a complete simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ ComplUSGraph)
Theorem	cplgr1vlem 29118	Lemma for cplgr1v 29119 and cusgr1v 29120. (Contributed by AV, 23-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((♯‘𝑉) = 1 → 𝐺 ∈ V)
Theorem	cplgr1v 29119	A graph with one vertex is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 14-Feb-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((♯‘𝑉) = 1 → 𝐺 ∈ ComplGraph)
Theorem	cusgr1v 29120	A graph with one vertex and no edges is a complete simple graph. (Contributed by AV, 1-Nov-2020.) (Revised by AV, 23-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((♯‘𝑉) = 1 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ ComplUSGraph)
Theorem	cplgr2v 29121	An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝐺 ∈ ComplGraph ↔ 𝑉 ∈ 𝐸))
Theorem	cplgr2vpr 29122	An undirected hypergraph with two (different) vertices is complete iff 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.) (Revised by AV, 3-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝐺 ∈ UHGraph ∧ 𝑉 = {𝐴, 𝐵})) → (𝐺 ∈ ComplGraph ↔ {𝐴, 𝐵} ∈ 𝐸))
Theorem	nbcplgr 29123	In a complete graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
Theorem	cplgr3v 29124	A pseudograph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}    ⇒   ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
Theorem	cusgr3vnbpr 29125*	The neighbors of a vertex in a simple 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.) (Revised by AV, 5-Nov-2020.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ USGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ ComplGraph ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
Theorem	cplgrop 29126	A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
⊢ (𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)
Theorem	cusgrop 29127	A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
⊢ (𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
Theorem	cusgrexilem1 29128*	Lemma 1 for cusgrexi 29132. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}    ⇒   ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V)
Theorem	usgrexilem 29129*	Lemma for usgrexi 29130. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}    ⇒   ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
Theorem	usgrexi 29130*	An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 10-Nov-2021.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}    ⇒   ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph)
Theorem	cusgrexilem2 29131*	Lemma 2 for cusgrexi 29132. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}    ⇒   ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)
Theorem	cusgrexi 29132*	An arbitrary set 𝑉 regarded as set of vertices together with the set of pairs of elements of this set regarded as edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 14-Feb-2022.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}    ⇒   ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)
Theorem	cusgrexg 29133*	For each set there is a set of edges so that the set together with these edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
⊢ (𝑉 ∈ 𝑊 → ∃𝑒⟨𝑉, 𝑒⟩ ∈ ComplUSGraph)
Theorem	structtousgr 29134*	Any (extensible) structure with a base set can be made a simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘𝑆) ∣ (♯‘𝑥) = 2}    &   ⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ 𝐺 = (𝑆 sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)    ⇒   ⊢ (𝜑 → 𝐺 ∈ USGraph)
Theorem	structtocusgr 29135*	Any (extensible) structure with a base set can be made a complete simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.) (Proof shortened by AV, 14-Feb-2022.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘𝑆) ∣ (♯‘𝑥) = 2}    &   ⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ 𝐺 = (𝑆 sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)    ⇒   ⊢ (𝜑 → 𝐺 ∈ ComplUSGraph)
Theorem	cffldtocusgr 29136*	The field of complex numbers can be made a complete simple graph with the set of pairs of complex numbers regarded as edges. This theorem demonstrates the capabilities of the current definitions for graphs applied to extensible structures. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.)
⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2}    &   ⊢ 𝐺 = (ℂfld sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)    ⇒   ⊢ 𝐺 ∈ ComplUSGraph
Theorem	cusgrres 29137*	Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}    &   ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph)
Theorem	cusgrsizeindb0 29138	Base case of the induction in cusgrsize 29143. The size of a complete simple graph with 0 vertices, actually of every null graph, is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = ((♯‘𝑉)C2))
Theorem	cusgrsizeindb1 29139	Base case of the induction in cusgrsize 29143. 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.) (Revised by AV, 7-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = ((♯‘𝑉)C2))
Theorem	cusgrsizeindslem 29140*	Lemma for cusgrsizeinds 29141. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((♯‘𝑉) − 1))
Theorem	cusgrsizeinds 29141*	Part 1 of induction step in cusgrsize 29143. The size of a complete simple graph with 𝑛 vertices is (𝑛 − 1) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘𝐸) = (((♯‘𝑉) − 1) + (♯‘𝐹)))
Theorem	cusgrsize2inds 29142*	Induction step in cusgrsize 29143. If the size of the complete graph with 𝑛 vertices reduced by one vertex is "(𝑛 − 1) choose 2", the size of the complete graph with 𝑛 vertices is "𝑛 choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}    ⇒   ⊢ (𝑌 ∈ ℕ0 → ((𝐺 ∈ ComplUSGraph ∧ (♯‘𝑉) = 𝑌 ∧ 𝑁 ∈ 𝑉) → ((♯‘𝐹) = ((♯‘(𝑉 ∖ {𝑁}))C2) → (♯‘𝐸) = ((♯‘𝑉)C2))))
Theorem	cusgrsize 29143	The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = ((♯‘𝑉)C2))
Theorem	cusgrfilem1 29144*	Lemma 1 for cusgrfi 29147. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺))
Theorem	cusgrfilem2 29145*	Lemma 2 for cusgrfi 29147. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}    &   ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃)
Theorem	cusgrfilem3 29146*	Lemma 3 for cusgrfi 29147. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}    &   ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝑉 ∈ Fin ↔ 𝑃 ∈ Fin))
Theorem	cusgrfi 29147	If the size of a complete simple graph is finite, then its order is also finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin)
Theorem	usgredgsscusgredg 29148	A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → 𝐸 ⊆ 𝐹)
Theorem	usgrsscusgr 29149*	A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ∀𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 𝑒 = 𝑓)
Theorem	sizusglecusglem1 29150	Lemma 1 for sizusglecusg 29152. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ( I ↾ 𝐸):𝐸–1-1→𝐹)
Theorem	sizusglecusglem2 29151	Lemma 2 for sizusglecusg 29152. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ∧ 𝐹 ∈ Fin) → 𝐸 ∈ Fin)
Theorem	sizusglecusg 29152	The size of a simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐻)    &   ⊢ 𝐹 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹))
Theorem	fusgrmaxsize 29153	The maximum size of a finite simple graph with 𝑛 vertices is (((𝑛 − 1)∗𝑛) / 2). See statement in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 14-Nov-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
17.2.10  Vertex degree
Syntax	cvtxdg 29154	Extend class notation with the vertex degree function.
class VtxDeg
Definition	df-vtxdg 29155*	Define the vertex degree function for a graph. To be appropriate for arbitrary hypergraphs, we have to double-count those edges that contain 𝑢 "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is not of finite size), the extended addition +𝑒 is used for the summation of the number of "ordinary" edges" and the number of "loops". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
⊢ VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))
Theorem	vtxdgfval 29156*	The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = (𝑢 ∈ 𝑉 ↦ ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑢}}))))
Theorem	vtxdgval 29157*	The degree of a vertex. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
Theorem	vtxdgfival 29158*	The degree of a vertex for graphs of finite size. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 8-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}) + (♯‘{𝑥 ∈ 𝐴 ∣ (𝐼‘𝑥) = {𝑈}})))
Theorem	vtxdgop 29159	The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)
⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
Theorem	vtxdgf 29160	The vertex degree function is a function from vertices to extended nonnegative integers. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ0*)
Theorem	vtxdgelxnn0 29161	The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑋) ∈ ℕ0*)
Theorem	vtxdg0v 29162	The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 = ∅ ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Theorem	vtxdg0e 29163	The degree of a vertex in an empty graph is zero, because there are no edges. This is the base case for the induction for calculating the degree of a vertex, for example in a Königsberg graph (see also the induction steps vdegp1ai 29225, vdegp1bi 29226 and vdegp1ci 29227). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Theorem	vtxdgfisnn0 29164	The degree of a vertex in a graph of finite size is a nonnegative integer. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ ((𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0)
Theorem	vtxdgfisf 29165	The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ0)
Theorem	vtxdeqd 29166	Equality theorem for the vertex degree: If two graphs are structurally equal, their vertex degree functions are equal. (Contributed by AV, 26-Feb-2021.)
⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑌)    &   ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺))    &   ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))    ⇒   ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))
Theorem	vtxduhgr0e 29167	The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 29163. (Contributed by AV, 15-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉 ∧ 𝐸 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Theorem	vtxdlfuhgr1v 29168*	The degree of the vertex in a loop-free hypergraph with one vertex is 0. (Contributed by AV, 2-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0))
Theorem	vdumgr0 29169	A vertex in a multigraph has degree 0 if the graph consists of only one vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 2-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉 ∧ (♯‘𝑉) = 1) → ((VtxDeg‘𝐺)‘𝑁) = 0)
Theorem	vtxdun 29170	The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 19-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → Fun 𝐽)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))
Theorem	vtxdfiun 29171	The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 19-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → Fun 𝐽)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))    &   ⊢ (𝜑 → dom 𝐼 ∈ Fin)    &   ⊢ (𝜑 → dom 𝐽 ∈ Fin)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))
Theorem	vtxduhgrun 29172	The degree of a vertex in the union of two hypergraphs on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   ⊢ (𝜑 → 𝐺 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ UHGraph)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))
Theorem	vtxduhgrfiun 29173	The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 7-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐽 = (iEdg‘𝐻)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)    &   ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   ⊢ (𝜑 → 𝐺 ∈ UHGraph)    &   ⊢ (𝜑 → 𝐻 ∈ UHGraph)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))    &   ⊢ (𝜑 → dom 𝐼 ∈ Fin)    &   ⊢ (𝜑 → dom 𝐽 ∈ Fin)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))
Theorem	vtxdlfgrval 29174*	The value of the vertex degree function for a loop-free graph 𝐺. (Contributed by AV, 23-Feb-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
Theorem	vtxdumgrval 29175*	The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 23-Feb-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
Theorem	vtxdusgrval 29176*	The value of the vertex degree function for a simple graph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐴 = dom 𝐼    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐼‘𝑥)}))
Theorem	vtxd0nedgb 29177*	A vertex has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
Theorem	vtxdushgrfvedglem 29178*	Lemma for vtxdushgrfvedg 29179 and vtxdusgrfvedg 29180. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉) → (♯‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
Theorem	vtxdushgrfvedg 29179*	The value of the vertex degree function for a simple hypergraph. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) +𝑒 (♯‘{𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑈}})))
Theorem	vtxdusgrfvedg 29180*	The value of the vertex degree function for a simple graph. (Contributed by AV, 12-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}))
Theorem	vtxduhgr0nedg 29181*	If a vertex in a hypergraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉 ∧ (𝐷‘𝑈) = 0) → ¬ ∃𝑣 ∈ 𝑉 {𝑈, 𝑣} ∈ 𝐸)
Theorem	vtxdumgr0nedg 29182*	If a vertex in a multigraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 15-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉 ∧ (𝐷‘𝑈) = 0) → ¬ ∃𝑣 ∈ 𝑉 {𝑈, 𝑣} ∈ 𝐸)
Theorem	vtxduhgr0edgnel 29183*	A vertex in a hypergraph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 24-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))
Theorem	vtxdusgr0edgnel 29184*	A vertex in a simple graph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))
Theorem	vtxdusgr0edgnelALT 29185*	Alternate proof of vtxdusgr0edgnel 29184, not based on vtxduhgr0edgnel 29183. A vertex in a simple graph has degree 0 if there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))
Theorem	vtxdgfusgrf 29186	The vertex degree function on finite simple graphs is a function from vertices to nonnegative integers. (Contributed by AV, 12-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → (VtxDeg‘𝐺):𝑉⟶ℕ0)
Theorem	vtxdgfusgr 29187*	In a finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 12-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
Theorem	fusgrn0degnn0 29188*	In a nonempty, finite graph there is a vertex having a nonnegative integer as degree. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 1-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑣 ∈ 𝑉 ∃𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)
Theorem	1loopgruspgr 29189	A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 28938). (Contributed by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    ⇒   ⊢ (𝜑 → 𝐺 ∈ USPGraph)
Theorem	1loopgredg 29190	The set of edges in a graph (simple pseudograph) with one edge which is a loop is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    ⇒   ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}})
Theorem	1loopgrnb0 29191	In a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    ⇒   ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅)
Theorem	1loopgrvd2 29192	The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = 2)
Theorem	1loopgrvd0 29193	The vertex degree of a one-edge graph, case 1 (for a loop): a loop at a vertex other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    &   ⊢ (𝜑 → 𝐾 ∈ (𝑉 ∖ {𝑁}))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐾) = 0)
Theorem	1hevtxdg0 29194	The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 is 0 if 𝐷 is not incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.)
⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐷 ∉ 𝐸)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0)
Theorem	1hevtxdg1 29195	The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 (not being a loop) is 1 if 𝐷 is incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)
⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐸)    &   ⊢ (𝜑 → 2 ≤ (♯‘𝐸))    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 1)
Theorem	1hegrvtxdg1 29196	The vertex degree of a graph with one hyperedge, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐵) = 1)
Theorem	1hegrvtxdg1r 29197	The vertex degree of a graph with one hyperedge, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1)
Theorem	1egrvtxdg1 29198	The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐵) = 1)
Theorem	1egrvtxdg1r 29199	The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1)
Theorem	1egrvtxdg0 29200	The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐷)    &   ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐷}⟩})    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 0)