Theorem elopaelxp

Description: Membership in an ordered-pair class abstraction implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018)

	Ref	Expression
Assertion	elopaelxp	$$⊢A\in \left\{⟨x,y⟩|\psi \right\}\to A\in \left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(A=⟨x,y⟩\wedge \psi \right)\to A=⟨x,y⟩$$
2	1	2eximi	$$⊢\exists x\exists y\left(A=⟨x,y⟩\wedge \psi \right)\to \exists x\exists yA=⟨x,y⟩$$
3		elopab	$$⊢A\in \left\{⟨x,y⟩|\psi \right\}\leftrightarrow \exists x\exists y\left(A=⟨x,y⟩\wedge \psi \right)$$
4		elvv	$$⊢A\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \exists x\exists yA=⟨x,y⟩$$
5	2 3 4	3imtr4i	$$⊢A\in \left\{⟨x,y⟩|\psi \right\}\to A\in \left(\mathrm{V}\times \mathrm{V}\right)$$