Theorem dmsn0el

Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008)

	Ref	Expression
Assertion	dmsn0el	⊢ ( ∅ ∈ 𝐴 → dom { 𝐴 } = ∅ )

Proof

Step	Hyp	Ref	Expression
1		dmsnn0	⊢ ( 𝐴 ∈ ( V × V ) ↔ dom { 𝐴 } ≠ ∅ )
2		0nelelxp	⊢ ( 𝐴 ∈ ( V × V ) → ¬ ∅ ∈ 𝐴 )
3	1 2	sylbir	⊢ ( dom { 𝐴 } ≠ ∅ → ¬ ∅ ∈ 𝐴 )
4	3	necon4ai	⊢ ( ∅ ∈ 𝐴 → dom { 𝐴 } = ∅ )