Metamath Proof Explorer


Theorem iscplgr

Description: The property of being a complete graph. (Contributed by AV, 1-Nov-2020)

Ref Expression
Hypothesis cplgruvtxb.v 𝑉 = ( Vtx ‘ 𝐺 )
Assertion iscplgr ( 𝐺𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 cplgruvtxb.v 𝑉 = ( Vtx ‘ 𝐺 )
2 1 cplgruvtxb ( 𝐺𝑊 → ( 𝐺 ∈ ComplGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) )
3 eqss ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) )
4 1 uvtxssvtx ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉
5 dfss3 ( 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) )
6 5 anbi2i ( ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) ↔ ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 ∧ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
7 4 6 mpbiran ( ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) )
8 3 7 bitri ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) )
9 2 8 syl6bb ( 𝐺𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )