Theorem nbgrnself

Description: A vertex in a graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 3-Nov-2020) (Revised by AV, 21-Mar-2021)

	Ref	Expression
Hypothesis	nbgrnself.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	nbgrnself	⊢ ∀ 𝑣 ∈ 𝑉 𝑣 ∉ ( 𝐺 NeighbVtx 𝑣 )

Proof

Step	Hyp	Ref	Expression
1		nbgrnself.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		neldifsnd	⊢ ( 𝑣 ∈ 𝑉 → ¬ 𝑣 ∈ ( 𝑉 ∖ { 𝑣 } ) )
3	2	intnanrd	⊢ ( 𝑣 ∈ 𝑉 → ¬ ( 𝑣 ∈ ( 𝑉 ∖ { 𝑣 } ) ∧ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑣 } ⊆ 𝑒 ) )
4		df-nel	⊢ ( 𝑣 ∉ { 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 } ↔ ¬ 𝑣 ∈ { 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 } )
5		preq2	⊢ ( 𝑛 = 𝑣 → { 𝑣 , 𝑛 } = { 𝑣 , 𝑣 } )
6	5	sseq1d	⊢ ( 𝑛 = 𝑣 → ( { 𝑣 , 𝑛 } ⊆ 𝑒 ↔ { 𝑣 , 𝑣 } ⊆ 𝑒 ) )
7	6	rexbidv	⊢ ( 𝑛 = 𝑣 → ( ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑣 } ⊆ 𝑒 ) )
8	7	elrab	⊢ ( 𝑣 ∈ { 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 } ↔ ( 𝑣 ∈ ( 𝑉 ∖ { 𝑣 } ) ∧ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑣 } ⊆ 𝑒 ) )
9	4 8	xchbinx	⊢ ( 𝑣 ∉ { 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 } ↔ ¬ ( 𝑣 ∈ ( 𝑉 ∖ { 𝑣 } ) ∧ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑣 } ⊆ 𝑒 ) )
10	3 9	sylibr	⊢ ( 𝑣 ∈ 𝑉 → 𝑣 ∉ { 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 } )
11		eqidd	⊢ ( 𝑣 ∈ 𝑉 → 𝑣 = 𝑣 )
12		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
13	1 12	nbgrval	⊢ ( 𝑣 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑣 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 } )
14	11 13	neleq12d	⊢ ( 𝑣 ∈ 𝑉 → ( 𝑣 ∉ ( 𝐺 NeighbVtx 𝑣 ) ↔ 𝑣 ∉ { 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 } ) )
15	10 14	mpbird	⊢ ( 𝑣 ∈ 𝑉 → 𝑣 ∉ ( 𝐺 NeighbVtx 𝑣 ) )
16	15	rgen	⊢ ∀ 𝑣 ∈ 𝑉 𝑣 ∉ ( 𝐺 NeighbVtx 𝑣 )