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Metamath Proof Explorer

Theorem uvtxel1

Description: Characterization of a universal vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 14-Feb-2022)

Ref Expression
Hypotheses uvtxel.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
isuvtx.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion uvtxel1 ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑁 ∈ 𝑉 ∧ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑁 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑁 } ⊆ 𝑒 ) )

Proof

Step Hyp Ref Expression
1 uvtxel.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 isuvtx.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
3 sneq ⊢ ( 𝑛 = 𝑁 → { 𝑛 } = { 𝑁 } )
4 3 difeq2d ⊢ ( 𝑛 = 𝑁 → ( 𝑉 ∖ { 𝑛 } ) = ( 𝑉 ∖ { 𝑁 } ) )
5 preq2 ⊢ ( 𝑛 = 𝑁 → { 𝑘 , 𝑛 } = { 𝑘 , 𝑁 } )
6 5 sseq1d ⊢ ( 𝑛 = 𝑁 → ( { 𝑘 , 𝑛 } ⊆ 𝑒 ↔ { 𝑘 , 𝑁 } ⊆ 𝑒 ) )
7 6 rexbidv ⊢ ( 𝑛 = 𝑁 → ( ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑁 } ⊆ 𝑒 ) )
8 4 7 raleqbidv ⊢ ( 𝑛 = 𝑁 → ( ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑛 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑛 } ⊆ 𝑒 ↔ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑁 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑁 } ⊆ 𝑒 ) )
9 1 2 isuvtx ⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑛 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑛 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑛 } ⊆ 𝑒 }
10 8 9 elrab2 ⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑁 ∈ 𝑉 ∧ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑁 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑁 } ⊆ 𝑒 ) )