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Metamath Proof Explorer

Theorem elin2

Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015)

		Ref	Expression
	Hypothesis	elin2.x	$⊢ X = B \cap C$
	Assertion	elin2	$⊢ A \in X \leftrightarrow (A \in B \land A \in C)$

Step	Hyp	Ref	Expression
1		elin2.x	$⊢ X = B \cap C$
2	1	eleq2i	$⊢ A \in X \leftrightarrow A \in (B \cap C)$
3		elin	$⊢ A \in (B \cap C) \leftrightarrow (A \in B \land A \in C)$
4	2 3	bitri	$⊢ A \in X \leftrightarrow (A \in B \land A \in C)$