Metamath Proof Explorer

Theorem elelb

Description: Equivalence between two common ways to characterize elements of a class B : the LHS says that sets are elements of B if and only if they satisfy ph while the RHS says that classes are elements of B if and only if they are sets and satisfy ph . Therefore, the LHS is a characterization among sets while the RHS is a characterization among classes. Note that the LHS is often formulated using a class variable instead of the universe _V while this is not possible for the RHS (apart from using B itself, which would not be very useful). (Contributed by BJ, 26-Feb-2023)

Ref Expression
Assertion elelb ⊢ A ∈ V → A ∈ B ↔ φ ↔ A ∈ B ↔ A ∈ V ∧ φ

Proof

Step Hyp Ref Expression
1 elex ⊢ A ∈ B → A ∈ V
2 1 biadani ⊢ A ∈ V → A ∈ B ↔ φ ↔ A ∈ B ↔ A ∈ V ∧ φ