Theorem frgrusgr

Description: A friendship graph is a simple graph. (Contributed by Alexander van der Vekens, 4-Oct-2017) (Revised by AV, 29-Mar-2021)

	Ref	Expression
Assertion	frgrusgr	$$⊢G\in \mathrm{FriendGraph}\to G\in \mathrm{USGraph}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(G\right)$$
2		eqid	$$⊢\mathrm{Edg}\left(G\right)=\mathrm{Edg}\left(G\right)$$
3	1 2	isfrgr	$$⊢G\in \mathrm{FriendGraph}\leftrightarrow \left(G\in \mathrm{USGraph}\wedge \forall k\in \mathrm{Vtx}\left(G\right)\forall l\in \left(\mathrm{Vtx}\left(G\right)\setminus \left\{k\right\}\right)\exists !x\in \mathrm{Vtx}\left(G\right)\left\{\left\{x,k\right\},\left\{x,l\right\}\right\}\subseteq \mathrm{Edg}\left(G\right)\right)$$
4	3	simplbi	$$⊢G\in \mathrm{FriendGraph}\to G\in \mathrm{USGraph}$$