Theorem clel2

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

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

Proof

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