Theorem upgredgssspr

Description: The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021)

	Ref	Expression
Assertion	upgredgssspr	$$⊢G\in \mathrm{UPGraph}\to \mathrm{Edg}\left(G\right)\subseteq \mathrm{Pairs}\left(\mathrm{Vtx}\left(G\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		upgredgss	$$⊢G\in \mathrm{UPGraph}\to \mathrm{Edg}\left(G\right)\subseteq \left\{p\in \left(𝒫\mathrm{Vtx}\left(G\right)\setminus \left\{\varnothing \right\}\right)|\left|p\right|\le 2\right\}$$
2		fvex	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{V}$$
3		sprvalpwle2	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{V}\to \mathrm{Pairs}\left(\mathrm{Vtx}\left(G\right)\right)=\left\{p\in \left(𝒫\mathrm{Vtx}\left(G\right)\setminus \left\{\varnothing \right\}\right)|\left|p\right|\le 2\right\}$$
4	2 3	ax-mp	$$⊢\mathrm{Pairs}\left(\mathrm{Vtx}\left(G\right)\right)=\left\{p\in \left(𝒫\mathrm{Vtx}\left(G\right)\setminus \left\{\varnothing \right\}\right)|\left|p\right|\le 2\right\}$$
5	1 4	sseqtrrdi	$$⊢G\in \mathrm{UPGraph}\to \mathrm{Edg}\left(G\right)\subseteq \mathrm{Pairs}\left(\mathrm{Vtx}\left(G\right)\right)$$