Theorem uhgredgss

Description: The set of edges of a hypergraph is a subset of the power set of vertices without the empty set. (Contributed by AV, 29-Nov-2020)

	Ref	Expression
Assertion	uhgredgss	$$⊢G\in \mathrm{UHGraph}\to \mathrm{Edg}\left(G\right)\subseteq 𝒫\mathrm{Vtx}\left(G\right)\setminus \left\{\varnothing \right\}$$

Proof

Step	Hyp	Ref	Expression
1		uhgredgn0	$$⊢\left(G\in \mathrm{UHGraph}\wedge x\in \mathrm{Edg}\left(G\right)\right)\to x\in \left(𝒫\mathrm{Vtx}\left(G\right)\setminus \left\{\varnothing \right\}\right)$$
2	1	ex	$$⊢G\in \mathrm{UHGraph}\to \left(x\in \mathrm{Edg}\left(G\right)\to x\in \left(𝒫\mathrm{Vtx}\left(G\right)\setminus \left\{\varnothing \right\}\right)\right)$$
3	2	ssrdv	$$⊢G\in \mathrm{UHGraph}\to \mathrm{Edg}\left(G\right)\subseteq 𝒫\mathrm{Vtx}\left(G\right)\setminus \left\{\varnothing \right\}$$