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Metamath Proof Explorer

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 ) ) )