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Metamath Proof Explorer

Theorem edgssv2

Description: An edge of a simple graph is a proper unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020) (Revised by AV, 23-Oct-2020)

Ref Expression
Hypotheses edgssv2.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
edgssv2.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion edgssv2 ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ⊆ 𝑉 ∧ ( ♯ ‘ 𝐶 ) = 2 ) )

Proof

Step Hyp Ref Expression
1 edgssv2.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 edgssv2.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
3 2 eleq2i ⊢ ( 𝐶 ∈ 𝐸 ↔ 𝐶 ∈ ( Edg ‘ 𝐺 ) )
4 edgusgr ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
5 3 4 sylan2b ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
6 elpwi ⊢ ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) → 𝐶 ⊆ ( Vtx ‘ 𝐺 ) )
7 6 anim1i ⊢ ( ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) → ( 𝐶 ⊆ ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
8 5 7 syl ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ⊆ ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
9 1 a1i ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → 𝑉 = ( Vtx ‘ 𝐺 ) )
10 9 sseq2d ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ⊆ 𝑉 ↔ 𝐶 ⊆ ( Vtx ‘ 𝐺 ) ) )
11 10 anbi1d ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( ( 𝐶 ⊆ 𝑉 ∧ ( ♯ ‘ 𝐶 ) = 2 ) ↔ ( 𝐶 ⊆ ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) ) )
12 8 11 mpbird ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ⊆ 𝑉 ∧ ( ♯ ‘ 𝐶 ) = 2 ) )