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Theorem elelb 35068
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 → (𝐴𝐵𝜑)) ↔ (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜑)))

Proof of Theorem elelb
StepHypRef Expression
1 elex 3448 . 2 (𝐴𝐵𝐴 ∈ V)
21biadani 817 1 ((𝐴 ∈ V → (𝐴𝐵𝜑)) ↔ (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3432
This theorem is referenced by: (None)
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