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Metamath Proof Explorer

Theorem 0fv

Description: Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014)

		Ref	Expression
	Assertion	0fv	$⊢ \emptyset (A) = \emptyset$

Step	Hyp	Ref	Expression
1		noel	$⊢ \neg A \in \emptyset$
2		dm0	$⊢ dom \emptyset = \emptyset$
3	2	eleq2i	$⊢ A \in dom \emptyset \leftrightarrow A \in \emptyset$
4	1 3	mtbir	$⊢ \neg A \in dom \emptyset$
5		ndmfv	$⊢ \neg A \in dom \emptyset \to \emptyset (A) = \emptyset$
6	4 5	ax-mp	$⊢ \emptyset (A) = \emptyset$