Theorem uvtxnm1nbgr

Description: A universal vertex has n - 1 neighbors in a finite graph with n vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017) (Revised by AV, 3-Nov-2020)

	Ref	Expression
Hypothesis	uvtxnm1nbgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	uvtxnm1nbgr	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) → ( ♯ ‘ ( 𝐺 NeighbVtx 𝑁 ) ) = ( ( ♯ ‘ 𝑉 ) − 1 ) )

Proof

Step	Hyp	Ref	Expression
1		uvtxnm1nbgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	uvtxnbgr	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )
3	2	adantl	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )
4	3	fveq2d	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) → ( ♯ ‘ ( 𝐺 NeighbVtx 𝑁 ) ) = ( ♯ ‘ ( 𝑉 ∖ { 𝑁 } ) ) )
5	1	fusgrvtxfi	⊢ ( 𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin )
6	1	uvtxisvtx	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → 𝑁 ∈ 𝑉 )
7	6	snssd	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → { 𝑁 } ⊆ 𝑉 )
8		hashssdif	⊢ ( ( 𝑉 ∈ Fin ∧ { 𝑁 } ⊆ 𝑉 ) → ( ♯ ‘ ( 𝑉 ∖ { 𝑁 } ) ) = ( ( ♯ ‘ 𝑉 ) − ( ♯ ‘ { 𝑁 } ) ) )
9	5 7 8	syl2an	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) → ( ♯ ‘ ( 𝑉 ∖ { 𝑁 } ) ) = ( ( ♯ ‘ 𝑉 ) − ( ♯ ‘ { 𝑁 } ) ) )
10		hashsng	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) → ( ♯ ‘ { 𝑁 } ) = 1 )
11	10	adantl	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) → ( ♯ ‘ { 𝑁 } ) = 1 )
12	11	oveq2d	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) → ( ( ♯ ‘ 𝑉 ) − ( ♯ ‘ { 𝑁 } ) ) = ( ( ♯ ‘ 𝑉 ) − 1 ) )
13	4 9 12	3eqtrd	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ) → ( ♯ ‘ ( 𝐺 NeighbVtx 𝑁 ) ) = ( ( ♯ ‘ 𝑉 ) − 1 ) )