Theorem dm0

Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of Monk1 p. 36. (Contributed by NM, 4-Jul-1994) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	dm0	⊢ dom ∅ = ∅

Proof

Step	Hyp	Ref	Expression
1		noel	⊢ ¬ ⟨ 𝑥 , 𝑦 ⟩ ∈ ∅
2	1	nex	⊢ ¬ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ ∅
3		vex	⊢ 𝑥 ∈ V
4	3	eldm2	⊢ ( 𝑥 ∈ dom ∅ ↔ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ ∅ )
5	2 4	mtbir	⊢ ¬ 𝑥 ∈ dom ∅
6	5	nel0	⊢ dom ∅ = ∅