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 ∈ V
Assertion	epeli	⊢ ( 𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		epeli.1	⊢ 𝐵 ∈ V
2		epelg	⊢ ( 𝐵 ∈ V → ( 𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵 )