Theorem umgrn1cycl

Description: In a multigraph graph (with no loops!) there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017) (Revised by AV, 2-Feb-2021)

	Ref	Expression
Assertion	umgrn1cycl	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → ( ♯ ‘ 𝐹 ) ≠ 1 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
3	1 2	umgrislfupgr	⊢ ( 𝐺 ∈ UMGraph ↔ ( 𝐺 ∈ UPGraph ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) )
4	1 2	lfgrn1cycl	⊢ ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ≠ 1 ) )
5	3 4	simplbiim	⊢ ( 𝐺 ∈ UMGraph → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ≠ 1 ) )
6	5	imp	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → ( ♯ ‘ 𝐹 ) ≠ 1 )