Theorem clel4

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

	Ref	Expression
Hypothesis	clel4.1	$$⊢B\in \mathrm{V}$$
Assertion	clel4	$$⊢A\in B\leftrightarrow \forall x\left(x=B\to A\in x\right)$$

Proof

Step	Hyp	Ref	Expression
1		clel4.1	$$⊢B\in \mathrm{V}$$
2		eleq2	$$⊢x=B\to \left(A\in x\leftrightarrow A\in B\right)$$
3	1 2	ceqsalv	$$⊢\forall x\left(x=B\to A\in x\right)\leftrightarrow A\in B$$
4	3	bicomi	$$⊢A\in B\leftrightarrow \forall x\left(x=B\to A\in x\right)$$