Theorem elsn

Description: There is exactly one element in a singleton. Exercise 2 of TakeutiZaring p. 15. (Contributed by NM, 13-Sep-1995)

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

Proof

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