Theorem uvtxel

Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 29-Oct-2020) (Revised by AV, 14-Feb-2022)

	Ref	Expression
Hypothesis	uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	uvtxel	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑁 ∈ 𝑉 ∧ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		sneq	⊢ ( 𝑣 = 𝑁 → { 𝑣 } = { 𝑁 } )
3	2	difeq2d	⊢ ( 𝑣 = 𝑁 → ( 𝑉 ∖ { 𝑣 } ) = ( 𝑉 ∖ { 𝑁 } ) )
4		oveq2	⊢ ( 𝑣 = 𝑁 → ( 𝐺 NeighbVtx 𝑣 ) = ( 𝐺 NeighbVtx 𝑁 ) )
5	4	eleq2d	⊢ ( 𝑣 = 𝑁 → ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) ) )
6	3 5	raleqbidv	⊢ ( 𝑣 = 𝑁 → ( ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) ) )
7	1	uvtxval	⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) }
8	6 7	elrab2	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑁 ∈ 𝑉 ∧ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑁 ) ) )