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Metamath Proof Explorer

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	$⊢ dom \emptyset = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		noel	$⊢ \neg ⟨x, y⟩ \in \emptyset$
2	1	nex	$⊢ \neg \exists y ⟨x, y⟩ \in \emptyset$
3		vex	$⊢ x \in V$
4	3	eldm2	$⊢ x \in dom \emptyset \leftrightarrow \exists y ⟨x, y⟩ \in \emptyset$
5	2 4	mtbir	$⊢ \neg x \in dom \emptyset$
6	5	nel0	$⊢ dom \emptyset = \emptyset$