Theorem umgr2v2evtxel

Description: A vertex in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020)

	Ref	Expression
Hypothesis	umgr2v2evtx.g	$$⊢G=⟨V,\left\{⟨0,\left\{A,B\right\}⟩,⟨1,\left\{A,B\right\}⟩\right\}⟩$$
Assertion	umgr2v2evtxel	$$⊢\left(V\in W\wedge A\in V\right)\to A\in \mathrm{Vtx}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		umgr2v2evtx.g	$$⊢G=⟨V,\left\{⟨0,\left\{A,B\right\}⟩,⟨1,\left\{A,B\right\}⟩\right\}⟩$$
2	1	umgr2v2evtx	$$⊢V\in W\to \mathrm{Vtx}\left(G\right)=V$$
3		eqcom	$$⊢\mathrm{Vtx}\left(G\right)=V\leftrightarrow V=\mathrm{Vtx}\left(G\right)$$
4	3	biimpi	$$⊢\mathrm{Vtx}\left(G\right)=V\to V=\mathrm{Vtx}\left(G\right)$$
5	4	eleq2d	$$⊢\mathrm{Vtx}\left(G\right)=V\to \left(A\in V\leftrightarrow A\in \mathrm{Vtx}\left(G\right)\right)$$
6	5	biimpcd	$$⊢A\in V\to \left(\mathrm{Vtx}\left(G\right)=V\to A\in \mathrm{Vtx}\left(G\right)\right)$$
7	2 6	mpan9	$$⊢\left(V\in W\wedge A\in V\right)\to A\in \mathrm{Vtx}\left(G\right)$$