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Metamath Proof Explorer

Theorem clel4

Description: Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993)

Ref Expression
Hypothesis clel4.1 ⊢ B ∈ V
Assertion clel4 ⊢ A ∈ B ↔ ∀ x x = B → A ∈ x

Proof

Step Hyp Ref Expression
1 clel4.1 ⊢ B ∈ V
2 clel4g ⊢ B ∈ V → A ∈ B ↔ ∀ x x = B → A ∈ x
3 1 2 ax-mp ⊢ A ∈ B ↔ ∀ x x = B → A ∈ x