Theorem elopaba

Description: Membership in an ordered-pair class abstraction. (Contributed by NM, 25-Feb-2014) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypothesis	copsex2ga.1	$$⊢A=⟨x,y⟩\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	elopaba	$$⊢A\in \left\{⟨x,y⟩|\psi \right\}\leftrightarrow \left(A\in \left(\mathrm{V}\times \mathrm{V}\right)\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		copsex2ga.1	$$⊢A=⟨x,y⟩\to \left(\phi \leftrightarrow \psi \right)$$
2		elopab	$$⊢A\in \left\{⟨x,y⟩|\psi \right\}\leftrightarrow \exists x\exists y\left(A=⟨x,y⟩\wedge \psi \right)$$
3	1	copsex2gb	$$⊢\exists x\exists y\left(A=⟨x,y⟩\wedge \psi \right)\leftrightarrow \left(A\in \left(\mathrm{V}\times \mathrm{V}\right)\wedge \phi \right)$$
4	2 3	bitri	$$⊢A\in \left\{⟨x,y⟩|\psi \right\}\leftrightarrow \left(A\in \left(\mathrm{V}\times \mathrm{V}\right)\wedge \phi \right)$$