Theorem clel3

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

	Ref	Expression
Hypothesis	clel3.1	$$⊢B\in \mathrm{V}$$
Assertion	clel3	$$⊢A\in B\leftrightarrow \exists x\left(x=B\wedge A\in x\right)$$

Proof

Step	Hyp	Ref	Expression
1		clel3.1	$$⊢B\in \mathrm{V}$$
2		clel3g	$$⊢B\in \mathrm{V}\to \left(A\in B\leftrightarrow \exists x\left(x=B\wedge A\in x\right)\right)$$
3	1 2	ax-mp	$$⊢A\in B\leftrightarrow \exists x\left(x=B\wedge A\in x\right)$$