Theorem nelsn

Description: If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020) (Proof shortened by BJ, 4-May-2021)

	Ref	Expression
Assertion	nelsn	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ { 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		elsni	⊢ ( 𝐴 ∈ { 𝐵 } → 𝐴 = 𝐵 )
2	1	necon3ai	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ { 𝐵 } )