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Metamath Proof Explorer

Theorem usgredgppr

Description: An edge of a simple graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge). Analogue of usgredg2 . (Contributed by Alexander van der Vekens, 11-Aug-2017) (Revised by AV, 9-Jan-2020) (Revised by AV, 23-Oct-2020)

		Ref	Expression
	Hypothesis	usgredgppr.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
	Assertion	usgredgppr	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( ♯ ‘ 𝐶 ) = 2 )

Proof

Step	Hyp	Ref	Expression
1		usgredgppr.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
2	1	eleq2i	⊢ ( 𝐶 ∈ 𝐸 ↔ 𝐶 ∈ ( Edg ‘ 𝐺 ) )
3		edgusgr	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
4	2 3	sylan2b	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
5	4	simprd	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( ♯ ‘ 𝐶 ) = 2 )