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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 ∈ 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 ) ) )