Theorem nbgr0edg

Description: In an empty graph (with no edges), every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 26-Oct-2020) (Proof shortened by AV, 15-Nov-2020)

	Ref	Expression
Assertion	nbgr0edg	⊢ ( ( Edg ‘ 𝐺 ) = ∅ → ( 𝐺 NeighbVtx 𝐾 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		rzal	⊢ ( ( Edg ‘ 𝐺 ) = ∅ → ∀ 𝑒 ∈ ( Edg ‘ 𝐺 ) ¬ { 𝐾 , 𝑛 } ⊆ 𝑒 )
2		ralnex	⊢ ( ∀ 𝑒 ∈ ( Edg ‘ 𝐺 ) ¬ { 𝐾 , 𝑛 } ⊆ 𝑒 ↔ ¬ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝐾 , 𝑛 } ⊆ 𝑒 )
3	1 2	sylib	⊢ ( ( Edg ‘ 𝐺 ) = ∅ → ¬ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝐾 , 𝑛 } ⊆ 𝑒 )
4	3	ralrimivw	⊢ ( ( Edg ‘ 𝐺 ) = ∅ → ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝐾 } ) ¬ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝐾 , 𝑛 } ⊆ 𝑒 )
5	4	nbgr0vtxlem	⊢ ( ( Edg ‘ 𝐺 ) = ∅ → ( 𝐺 NeighbVtx 𝐾 ) = ∅ )