Theorem elsn2

Description: There is exactly one element in a singleton. Exercise 2 of TakeutiZaring p. 15. This variation requires only that B , rather than A , be a set. (Contributed by NM, 12-Jun-1994)

	Ref	Expression
Hypothesis	elsn2.1	⊢ 𝐵 ∈ V
Assertion	elsn2	⊢ ( 𝐴 ∈ { 𝐵 } ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elsn2.1	⊢ 𝐵 ∈ V
2		elsn2g	⊢ ( 𝐵 ∈ V → ( 𝐴 ∈ { 𝐵 } ↔ 𝐴 = 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ { 𝐵 } ↔ 𝐴 = 𝐵 )