Theorem usgr0e

Description: The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 25-Nov-2020)

	Ref	Expression
Hypotheses	usgr0e.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
	usgr0e.e	⊢ ( 𝜑 → ( iEdg ‘ 𝐺 ) = ∅ )
Assertion	usgr0e	⊢ ( 𝜑 → 𝐺 ∈ USGraph )

Proof

Step	Hyp	Ref	Expression
1		usgr0e.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
2		usgr0e.e	⊢ ( 𝜑 → ( iEdg ‘ 𝐺 ) = ∅ )
3	2	f10d	⊢ ( 𝜑 → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
4		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
5		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
6	4 5	isusgr	⊢ ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
7	1 6	syl	⊢ ( 𝜑 → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
8	3 7	mpbird	⊢ ( 𝜑 → 𝐺 ∈ USGraph )