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Theorem elelb 34231
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.)
Ref Expression
elelb ((𝐴 ∈ V → (𝐴𝐵𝜑)) ↔ (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜑)))

Proof of Theorem elelb
StepHypRef Expression
1 elex 3489 . 2 (𝐴𝐵𝐴 ∈ V)
21biadani 819 1 ((𝐴 ∈ V → (𝐴𝐵𝜑)) ↔ (𝐴𝐵 ↔ (𝐴 ∈ V ∧ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  Vcvv 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473
This theorem is referenced by: (None)
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