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Metamath Proof Explorer

Theorem clel2

Description: Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993)

		Ref	Expression
	Hypothesis	clel2.1	$⊢ A \in V$
	Assertion	clel2	$⊢ A \in B \leftrightarrow \forall x (x = A \to x \in B)$

Step	Hyp	Ref	Expression
1		clel2.1	$⊢ A \in V$
2		clel2g	$⊢ A \in V \to (A \in B \leftrightarrow \forall x (x = A \to x \in B))$
3	1 2	ax-mp	$⊢ A \in B \leftrightarrow \forall x (x = A \to x \in B)$