Theorem elsn2g

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, 28-Oct-2003)

	Ref	Expression
Assertion	elsn2g	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ { 𝐵 } ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elsni	⊢ ( 𝐴 ∈ { 𝐵 } → 𝐴 = 𝐵 )
2		snidg	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 ∈ { 𝐵 } )
3		eleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ { 𝐵 } ↔ 𝐵 ∈ { 𝐵 } ) )
4	2 3	syl5ibrcom	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 = 𝐵 → 𝐴 ∈ { 𝐵 } ) )
5	1 4	impbid2	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ { 𝐵 } ↔ 𝐴 = 𝐵 ) )