Metamath Proof Explorer


Theorem sprssspr

Description: The set of all unordered pairs over a given set V is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021)

Ref Expression
Assertion sprssspr ( Pairs ‘ 𝑉 ) ⊆ { 𝑝 ∣ ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } }

Proof

Step Hyp Ref Expression
1 sprval ( 𝑉 ∈ V → ( Pairs ‘ 𝑉 ) = { 𝑝 ∣ ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } } )
2 r2ex ( ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } ↔ ∃ 𝑎𝑏 ( ( 𝑎𝑉𝑏𝑉 ) ∧ 𝑝 = { 𝑎 , 𝑏 } ) )
3 simpr ( ( ( 𝑎𝑉𝑏𝑉 ) ∧ 𝑝 = { 𝑎 , 𝑏 } ) → 𝑝 = { 𝑎 , 𝑏 } )
4 3 2eximi ( ∃ 𝑎𝑏 ( ( 𝑎𝑉𝑏𝑉 ) ∧ 𝑝 = { 𝑎 , 𝑏 } ) → ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } )
5 2 4 sylbi ( ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } → ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } )
6 5 ax-gen 𝑝 ( ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } → ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } )
7 6 a1i ( 𝑉 ∈ V → ∀ 𝑝 ( ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } → ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } ) )
8 ss2ab ( { 𝑝 ∣ ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } } ⊆ { 𝑝 ∣ ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } } ↔ ∀ 𝑝 ( ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } → ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } ) )
9 7 8 sylibr ( 𝑉 ∈ V → { 𝑝 ∣ ∃ 𝑎𝑉𝑏𝑉 𝑝 = { 𝑎 , 𝑏 } } ⊆ { 𝑝 ∣ ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } } )
10 1 9 eqsstrd ( 𝑉 ∈ V → ( Pairs ‘ 𝑉 ) ⊆ { 𝑝 ∣ ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } } )
11 fvprc ( ¬ 𝑉 ∈ V → ( Pairs ‘ 𝑉 ) = ∅ )
12 0ss ∅ ⊆ { 𝑝 ∣ ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } }
13 12 a1i ( ¬ 𝑉 ∈ V → ∅ ⊆ { 𝑝 ∣ ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } } )
14 11 13 eqsstrd ( ¬ 𝑉 ∈ V → ( Pairs ‘ 𝑉 ) ⊆ { 𝑝 ∣ ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } } )
15 10 14 pm2.61i ( Pairs ‘ 𝑉 ) ⊆ { 𝑝 ∣ ∃ 𝑎𝑏 𝑝 = { 𝑎 , 𝑏 } }