Metamath Proof Explorer


Theorem usgredgleord

Description: In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018) (Revised by AV, 18-Oct-2020) (Proof shortened by AV, 6-Dec-2020)

Ref Expression
Hypotheses usgredgleord.v 𝑉 = ( Vtx ‘ 𝐺 )
usgredgleord.e 𝐸 = ( Edg ‘ 𝐺 )
Assertion usgredgleord ( ( 𝐺 ∈ USGraph ∧ 𝑁𝑉 ) → ( ♯ ‘ { 𝑒𝐸𝑁𝑒 } ) ≤ ( ♯ ‘ 𝑉 ) )

Proof

Step Hyp Ref Expression
1 usgredgleord.v 𝑉 = ( Vtx ‘ 𝐺 )
2 usgredgleord.e 𝐸 = ( Edg ‘ 𝐺 )
3 usgruspgr ( 𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
4 1 2 uspgredgleord ( ( 𝐺 ∈ USPGraph ∧ 𝑁𝑉 ) → ( ♯ ‘ { 𝑒𝐸𝑁𝑒 } ) ≤ ( ♯ ‘ 𝑉 ) )
5 3 4 sylan ( ( 𝐺 ∈ USGraph ∧ 𝑁𝑉 ) → ( ♯ ‘ { 𝑒𝐸𝑁𝑒 } ) ≤ ( ♯ ‘ 𝑉 ) )