Theorem prprreueq

Description: There is a unique proper unordered pair over a given set V fulfilling a wff iff there is a unique subset of V of size two fulfilling this wff. (Contributed by AV, 29-Apr-2023)

	Ref	Expression
Assertion	prprreueq	⊢ ( 𝑉 ∈ 𝑊 → ( ∃! 𝑝 ∈ ( Pairsproper ‘ 𝑉 ) 𝜑 ↔ ∃! 𝑝 ∈ 𝒫 𝑉 ( ( ♯ ‘ 𝑝 ) = 2 ∧ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		prprelb	⊢ ( 𝑉 ∈ 𝑊 → ( 𝑝 ∈ ( Pairsproper ‘ 𝑉 ) ↔ ( 𝑝 ∈ 𝒫 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 2 ) ) )
2	1	anbi1d	⊢ ( 𝑉 ∈ 𝑊 → ( ( 𝑝 ∈ ( Pairsproper ‘ 𝑉 ) ∧ 𝜑 ) ↔ ( ( 𝑝 ∈ 𝒫 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 2 ) ∧ 𝜑 ) ) )
3		anass	⊢ ( ( ( 𝑝 ∈ 𝒫 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 2 ) ∧ 𝜑 ) ↔ ( 𝑝 ∈ 𝒫 𝑉 ∧ ( ( ♯ ‘ 𝑝 ) = 2 ∧ 𝜑 ) ) )
4	2 3	bitrdi	⊢ ( 𝑉 ∈ 𝑊 → ( ( 𝑝 ∈ ( Pairsproper ‘ 𝑉 ) ∧ 𝜑 ) ↔ ( 𝑝 ∈ 𝒫 𝑉 ∧ ( ( ♯ ‘ 𝑝 ) = 2 ∧ 𝜑 ) ) ) )
5	4	eubidv	⊢ ( 𝑉 ∈ 𝑊 → ( ∃! 𝑝 ( 𝑝 ∈ ( Pairsproper ‘ 𝑉 ) ∧ 𝜑 ) ↔ ∃! 𝑝 ( 𝑝 ∈ 𝒫 𝑉 ∧ ( ( ♯ ‘ 𝑝 ) = 2 ∧ 𝜑 ) ) ) )
6		df-reu	⊢ ( ∃! 𝑝 ∈ ( Pairsproper ‘ 𝑉 ) 𝜑 ↔ ∃! 𝑝 ( 𝑝 ∈ ( Pairsproper ‘ 𝑉 ) ∧ 𝜑 ) )
7		df-reu	⊢ ( ∃! 𝑝 ∈ 𝒫 𝑉 ( ( ♯ ‘ 𝑝 ) = 2 ∧ 𝜑 ) ↔ ∃! 𝑝 ( 𝑝 ∈ 𝒫 𝑉 ∧ ( ( ♯ ‘ 𝑝 ) = 2 ∧ 𝜑 ) ) )
8	5 6 7	3bitr4g	⊢ ( 𝑉 ∈ 𝑊 → ( ∃! 𝑝 ∈ ( Pairsproper ‘ 𝑉 ) 𝜑 ↔ ∃! 𝑝 ∈ 𝒫 𝑉 ( ( ♯ ‘ 𝑝 ) = 2 ∧ 𝜑 ) ) )