Theorem uvtx0

Description: There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 30-Oct-2020) (Proof shortened by AV, 14-Feb-2022)

	Ref	Expression
Hypothesis	uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	uvtx0	⊢ ( 𝑉 = ∅ → ( UnivVtx ‘ 𝐺 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	uvtxval	⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) }
3		rabeq	⊢ ( 𝑉 = ∅ → { 𝑣 ∈ 𝑉 ∣ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) } = { 𝑣 ∈ ∅ ∣ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) } )
4		rab0	⊢ { 𝑣 ∈ ∅ ∣ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) } = ∅
5	3 4	eqtrdi	⊢ ( 𝑉 = ∅ → { 𝑣 ∈ 𝑉 ∣ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) } = ∅ )
6	2 5	eqtrid	⊢ ( 𝑉 = ∅ → ( UnivVtx ‘ 𝐺 ) = ∅ )