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Database GRAPH THEORY Undirected graphs Undirected simple graphs 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 } ) )