Metamath Proof Explorer


Theorem prsssprel

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

Ref Expression
Assertion prsssprel ( ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ∧ { 𝑋 , 𝑌 } ∈ 𝑃 ∧ ( 𝑋𝑈𝑌𝑊 ) ) → ( 𝑋𝑉𝑌𝑉 ) )

Proof

Step Hyp Ref Expression
1 ssel2 ( ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ∧ { 𝑋 , 𝑌 } ∈ 𝑃 ) → { 𝑋 , 𝑌 } ∈ ( Pairs ‘ 𝑉 ) )
2 prsprel ( ( { 𝑋 , 𝑌 } ∈ ( Pairs ‘ 𝑉 ) ∧ ( 𝑋𝑈𝑌𝑊 ) ) → ( 𝑋𝑉𝑌𝑉 ) )
3 1 2 stoic3 ( ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ∧ { 𝑋 , 𝑌 } ∈ 𝑃 ∧ ( 𝑋𝑈𝑌𝑊 ) ) → ( 𝑋𝑉𝑌𝑉 ) )