Metamath Proof Explorer

Theorem elvvOLD

Description: Obsolete version of elvv as of 16-Oct-2026. (Contributed by NM, 4-Jul-1994) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	elvvOLD	$⊢ A \in (V \times V) \leftrightarrow \exists x \exists y A = ⟨x, y⟩$

Proof

Step	Hyp	Ref	Expression
1		elxp	$⊢ A \in (V \times V) \leftrightarrow \exists x \exists y (A = ⟨x, y⟩ \land (x \in V \land y \in V))$
2		vex	$⊢ x \in V$
3		vex	$⊢ y \in V$
4	2 3	pm3.2i	$⊢ (x \in V \land y \in V)$
5	4	biantru	$⊢ A = ⟨x, y⟩ \leftrightarrow (A = ⟨x, y⟩ \land (x \in V \land y \in V))$
6	5	2exbii	$⊢ \exists x \exists y A = ⟨x, y⟩ \leftrightarrow \exists x \exists y (A = ⟨x, y⟩ \land (x \in V \land y \in V))$
7	1 6	bitr4i	$⊢ A \in (V \times V) \leftrightarrow \exists x \exists y A = ⟨x, y⟩$