Theorem epeli

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg . (Contributed by Scott Fenton, 11-Apr-2012)

	Ref	Expression
Hypothesis	epeli.1	$$⊢B\in \mathrm{V}$$
Assertion	epeli	$$⊢A\mathrm{E}B\leftrightarrow A\in B$$

Proof

Step	Hyp	Ref	Expression
1		epeli.1	$$⊢B\in \mathrm{V}$$
2		epelg	$$⊢B\in \mathrm{V}\to \left(A\mathrm{E}B\leftrightarrow A\in B\right)$$
3	1 2	ax-mp	$$⊢A\mathrm{E}B\leftrightarrow A\in B$$