Theorem usgrnloop0

Description: A simple graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

	Ref	Expression
Hypothesis	usgrnloopv.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	usgrnloop0	⊢ ( 𝐺 ∈ USGraph → { 𝑥 ∈ dom 𝐸 ∣ ( 𝐸 ‘ 𝑥 ) = { 𝑈 } } = ∅ )

Proof

Step	Hyp	Ref	Expression
1		usgrnloopv.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2		usgrumgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
3	1	umgrnloop0	⊢ ( 𝐺 ∈ UMGraph → { 𝑥 ∈ dom 𝐸 ∣ ( 𝐸 ‘ 𝑥 ) = { 𝑈 } } = ∅ )
4	2 3	syl	⊢ ( 𝐺 ∈ USGraph → { 𝑥 ∈ dom 𝐸 ∣ ( 𝐸 ‘ 𝑥 ) = { 𝑈 } } = ∅ )