Theorem edgusgr

Description: An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020) (Revised by AV, 14-Oct-2020)

	Ref	Expression
Assertion	edgusgr	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐸 ) = 2 ) )

Proof

Step	Hyp	Ref	Expression
1		usgredgss	⊢ ( 𝐺 ∈ USGraph → ( Edg ‘ 𝐺 ) ⊆ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
2	1	sselda	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → 𝐸 ∈ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
3		fveqeq2	⊢ ( 𝑥 = 𝐸 → ( ( ♯ ‘ 𝑥 ) = 2 ↔ ( ♯ ‘ 𝐸 ) = 2 ) )
4	3	elrab	⊢ ( 𝐸 ∈ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ↔ ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐸 ) = 2 ) )
5	2 4	sylib	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐸 ) = 2 ) )