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 ∈ FriendGraph → 𝐺 ∈ USGraph )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
3	1 2	isfrgr	⊢ ( 𝐺 ∈ FriendGraph ↔ ( 𝐺 ∈ USGraph ∧ ∀ 𝑘 ∈ ( Vtx ‘ 𝐺 ) ∀ 𝑙 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑘 } ) ∃! 𝑥 ∈ ( Vtx ‘ 𝐺 ) { { 𝑥 , 𝑘 } , { 𝑥 , 𝑙 } } ⊆ ( Edg ‘ 𝐺 ) ) )
4	3	simplbi	⊢ ( 𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )