Theorem vdiscusgrb

Description: A finite simple graph with n vertices is complete iff every vertex has degree n - 1 . (Contributed by Alexander van der Vekens, 14-Jul-2018) (Revised by AV, 22-Dec-2020)

	Ref	Expression
Hypothesis	hashnbusgrvd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	vdiscusgrb	⊢ ( 𝐺 ∈ FinUSGraph → ( 𝐺 ∈ ComplUSGraph ↔ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hashnbusgrvd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		fusgrusgr	⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
3	1	cusgruvtxb	⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) )
4	1	uvtxssvtx	⊢ ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉
5		eqcom	⊢ ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ 𝑉 = ( UnivVtx ‘ 𝐺 ) )
6		sssseq	⊢ ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 → ( 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ↔ 𝑉 = ( UnivVtx ‘ 𝐺 ) ) )
7	5 6	bitr4id	⊢ ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 → ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) )
8	4 7	mp1i	⊢ ( 𝐺 ∈ USGraph → ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) )
9	3 8	bitrd	⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) )
10		dfss3	⊢ ( 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) )
11	9 10	bitrdi	⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
12	2 11	syl	⊢ ( 𝐺 ∈ FinUSGraph → ( 𝐺 ∈ ComplUSGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
13	1	usgruvtxvdb	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉 ) → ( 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )
14	13	ralbidva	⊢ ( 𝐺 ∈ FinUSGraph → ( ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )
15	12 14	bitrd	⊢ ( 𝐺 ∈ FinUSGraph → ( 𝐺 ∈ ComplUSGraph ↔ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) )