Theorem uspgrloopvtxel

Description: A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop ). (Contributed by AV, 17-Dec-2020)

	Ref	Expression
Hypothesis	uspgrloopvtx.g	$$⊢G=⟨V,\left\{⟨A,\left\{N\right\}⟩\right\}⟩$$
Assertion	uspgrloopvtxel	$$⊢\left(V\in W\wedge N\in V\right)\to N\in \mathrm{Vtx}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		uspgrloopvtx.g	$$⊢G=⟨V,\left\{⟨A,\left\{N\right\}⟩\right\}⟩$$
2	1	uspgrloopvtx	$$⊢V\in W\to \mathrm{Vtx}\left(G\right)=V$$
3		eleq2	$$⊢V=\mathrm{Vtx}\left(G\right)\to \left(N\in V\leftrightarrow N\in \mathrm{Vtx}\left(G\right)\right)$$
4	3	biimpd	$$⊢V=\mathrm{Vtx}\left(G\right)\to \left(N\in V\to N\in \mathrm{Vtx}\left(G\right)\right)$$
5	4	eqcoms	$$⊢\mathrm{Vtx}\left(G\right)=V\to \left(N\in V\to N\in \mathrm{Vtx}\left(G\right)\right)$$
6	5	com12	$$⊢N\in V\to \left(\mathrm{Vtx}\left(G\right)=V\to N\in \mathrm{Vtx}\left(G\right)\right)$$
7	2 6	mpan9	$$⊢\left(V\in W\wedge N\in V\right)\to N\in \mathrm{Vtx}\left(G\right)$$