Theorem oppr

Description: Equality for ordered pairs implies equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023)

	Ref	Expression
Assertion	oppr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )

Proof

Step	Hyp	Ref	Expression
1		opthg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )
2		preq12	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } )
3	1 2	biimtrdi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )