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 ∧ ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 | ⊢ ( ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ∧ { 𝑋 , 𝑌 } ∈ 𝑃 ) → { 𝑋 , 𝑌 } ∈ ( Pairs ‘ 𝑉 ) ) | |
2 | prsprel | ⊢ ( ( { 𝑋 , 𝑌 } ∈ ( Pairs ‘ 𝑉 ) ∧ ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) | |
3 | 1 2 | stoic3 | ⊢ ( ( 𝑃 ⊆ ( Pairs ‘ 𝑉 ) ∧ { 𝑋 , 𝑌 } ∈ 𝑃 ∧ ( 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) |