Theorem usgredg

Description: For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017) (Revised by AV, 17-Oct-2020) (Shortened by AV, 25-Nov-2020.)

	Ref	Expression
Hypotheses	edgssv2.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	edgssv2.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	usgredg	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝐶 = { 𝑎 , 𝑏 } ) )

Proof

Step	Hyp	Ref	Expression
1		edgssv2.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		edgssv2.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		usgrumgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
4	1 2	umgredg	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐶 ∈ 𝐸 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝐶 = { 𝑎 , 𝑏 } ) )
5	3 4	sylan	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝐶 = { 𝑎 , 𝑏 } ) )