Theorem elelb 33401

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 𝜑 while the RHS says that classes are elements of 𝐵 if and only if they are sets and satisfy 𝜑. 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.)

Assertion

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

Proof of Theorem elelb

Step	Hyp	Ref	Expression
1		elex 3429	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2	1	biadani 855	1 ⊢ ((𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜑)) ↔ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2164  Vcvv 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1660  df-ex 1879  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416

This theorem is referenced by: (None)