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Metamath Proof Explorer

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