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Metamath Proof Explorer

Theorem elxp

Description: Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994)

		Ref	Expression
	Assertion	elxp	$⊢ A \in (B \times C) \leftrightarrow \exists x \exists y (A = ⟨x, y⟩ \land (x \in B \land y \in C))$

Step	Hyp	Ref	Expression
1		df-xp	$⊢ B \times C = \{⟨x, y⟩ \| (x \in B \land y \in C)\}$
2	1	eleq2i	$⊢ A \in (B \times C) \leftrightarrow A \in \{⟨x, y⟩ \| (x \in B \land y \in C)\}$
3		elopab	$⊢ A \in \{⟨x, y⟩ \| (x \in B \land y \in C)\} \leftrightarrow \exists x \exists y (A = ⟨x, y⟩ \land (x \in B \land y \in C))$
4	2 3	bitri	$⊢ A \in (B \times C) \leftrightarrow \exists x \exists y (A = ⟨x, y⟩ \land (x \in B \land y \in C))$