Theorem bj-elsnb

Description: Biconditional version of elsng . (Contributed by BJ, 18-Nov-2023)

	Ref	Expression
Assertion	bj-elsnb	⊢ ( 𝐴 ∈ { 𝐵 } ↔ ( 𝐴 ∈ V ∧ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ { 𝐵 } → 𝐴 ∈ V )
2		elsng	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ { 𝐵 } ↔ 𝐴 = 𝐵 ) )
3	1 2	biadanii	⊢ ( 𝐴 ∈ { 𝐵 } ↔ ( 𝐴 ∈ V ∧ 𝐴 = 𝐵 ) )