Theorem dfclel

Description: Characterization of the elements of a class. (Contributed by BJ, 27-Jun-2019)

	Ref	Expression
Assertion	dfclel	$$⊢A\in B\leftrightarrow \exists x\left(x=A\wedge x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		cleljust	$$⊢y\in z\leftrightarrow \exists u\left(u=y\wedge u\in z\right)$$
2		cleljust	$$⊢t\in t\leftrightarrow \exists v\left(v=t\wedge v\in t\right)$$
3	1 2	df-clel	$$⊢A\in B\leftrightarrow \exists x\left(x=A\wedge x\in B\right)$$