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Metamath Proof Explorer

Theorem iun0

Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

		Ref	Expression
	Assertion	iun0	$⊢ ⋃_{x \in A} \emptyset = \emptyset$

Step	Hyp	Ref	Expression
1		noel	$⊢ \neg y \in \emptyset$
2	1	a1i	$⊢ x \in A \to \neg y \in \emptyset$
3	2	nrex	$⊢ \neg \exists x \in A y \in \emptyset$
4		eliun	$⊢ y \in ⋃_{x \in A} \emptyset \leftrightarrow \exists x \in A y \in \emptyset$
5	3 4	mtbir	$⊢ \neg y \in ⋃_{x \in A} \emptyset$
6	5	nel0	$⊢ ⋃_{x \in A} \emptyset = \emptyset$