Theorem usgrn2cycl

Description: In a simple graph there are no cycles with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017) (Revised by AV, 4-Feb-2021)

	Ref	Expression
Assertion	usgrn2cycl	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → ( ♯ ‘ 𝐹 ) ≠ 2 )

Proof

Step	Hyp	Ref	Expression
1		usgruspgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
2		cycliscrct	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 )
3		uspgrn2crct	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 ) → ( ♯ ‘ 𝐹 ) ≠ 2 )
4	1 2 3	syl2an	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → ( ♯ ‘ 𝐹 ) ≠ 2 )