Metamath Proof Explorer


Theorem elopab

Description: Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998)

Ref Expression
Assertion elopab ( 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )

Proof

Step Hyp Ref Expression
1 elex ( 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } → 𝐴 ∈ V )
2 opex 𝑥 , 𝑦 ⟩ ∈ V
3 eleq1 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝐴 ∈ V ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ V ) )
4 2 3 mpbiri ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → 𝐴 ∈ V )
5 4 adantr ( ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) → 𝐴 ∈ V )
6 5 exlimivv ( ∃ 𝑥𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) → 𝐴 ∈ V )
7 elopabw ( 𝐴 ∈ V → ( 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
8 1 6 7 pm5.21nii ( 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )