Metamath Proof Explorer


Theorem sprel

Description: An element of the set of all unordered pairs over a given set V is a pair of elements of the set V . (Contributed by AV, 22-Nov-2021)

Ref Expression
Assertion sprel ( 𝑋 ∈ ( Pairs ‘ 𝑉 ) → ∃ 𝑎𝑉𝑏𝑉 𝑋 = { 𝑎 , 𝑏 } )

Proof

Step Hyp Ref Expression
1 elfvex ( 𝑋 ∈ ( Pairs ‘ 𝑉 ) → 𝑉 ∈ V )
2 sprvalpw ( 𝑉 ∈ V → ( Pairs ‘ 𝑉 ) = { 𝑝 ∈ 𝒫 𝑉 ∣ ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } } )
3 2 eleq2d ( 𝑉 ∈ V → ( 𝑋 ∈ ( Pairs ‘ 𝑉 ) ↔ 𝑋 ∈ { 𝑝 ∈ 𝒫 𝑉 ∣ ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } } ) )
4 eqeq1 ( 𝑝 = 𝑋 → ( 𝑝 = { 𝑎 , 𝑏 } ↔ 𝑋 = { 𝑎 , 𝑏 } ) )
5 4 2rexbidv ( 𝑝 = 𝑋 → ( ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } ↔ ∃ 𝑎𝑉𝑏𝑉 𝑋 = { 𝑎 , 𝑏 } ) )
6 5 elrab ( 𝑋 ∈ { 𝑝 ∈ 𝒫 𝑉 ∣ ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } } ↔ ( 𝑋 ∈ 𝒫 𝑉 ∧ ∃ 𝑎𝑉𝑏𝑉 𝑋 = { 𝑎 , 𝑏 } ) )
7 6 simprbi ( 𝑋 ∈ { 𝑝 ∈ 𝒫 𝑉 ∣ ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } } → ∃ 𝑎𝑉𝑏𝑉 𝑋 = { 𝑎 , 𝑏 } )
8 3 7 syl6bi ( 𝑉 ∈ V → ( 𝑋 ∈ ( Pairs ‘ 𝑉 ) → ∃ 𝑎𝑉𝑏𝑉 𝑋 = { 𝑎 , 𝑏 } ) )
9 1 8 mpcom ( 𝑋 ∈ ( Pairs ‘ 𝑉 ) → ∃ 𝑎𝑉𝑏𝑉 𝑋 = { 𝑎 , 𝑏 } )