Metamath Proof Explorer

Theorem clnbusgrfi

Description: The closed neighborhood of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by AV, 10-May-2025)

Ref Expression
Hypotheses clnbusgrf1o.v ⊢ V = Vtx ⁡ G
clnbusgrf1o.e ⊢ E = Edg ⁡ G
Assertion clnbusgrfi ⊢ G ∈ USGraph ∧ E ∈ Fin ∧ U ∈ V → G ClNeighbVtx U ∈ Fin

Proof

Step Hyp Ref Expression
1 clnbusgrf1o.v ⊢ V = Vtx ⁡ G
2 clnbusgrf1o.e ⊢ E = Edg ⁡ G
3 rabfi ⊢ E ∈ Fin → e ∈ E | U ∈ e ∈ Fin
4 3 3ad2ant2 ⊢ G ∈ USGraph ∧ E ∈ Fin ∧ U ∈ V → e ∈ E | U ∈ e ∈ Fin
5 1 2 edgusgrclnbfin ⊢ G ∈ USGraph ∧ U ∈ V → G ClNeighbVtx U ∈ Fin ↔ e ∈ E | U ∈ e ∈ Fin
6 5 3adant2 ⊢ G ∈ USGraph ∧ E ∈ Fin ∧ U ∈ V → G ClNeighbVtx U ∈ Fin ↔ e ∈ E | U ∈ e ∈ Fin
7 4 6 mpbird ⊢ G ∈ USGraph ∧ E ∈ Fin ∧ U ∈ V → G ClNeighbVtx U ∈ Fin