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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 e. _V -> ( A e. B <-> ph ) ) <-> ( A e. B <-> ( A e. _V /\ ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		elex	\|- ( A e. B -> A e. _V )
2	1	biadani	\|- ( ( A e. _V -> ( A e. B <-> ph ) ) <-> ( A e. B <-> ( A e. _V /\ ph ) ) )