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Metamath Proof Explorer

Theorem iunxsn

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004) (Proof shortened by Mario Carneiro, 25-Jun-2016)

	Ref	Expression
Hypotheses	iunxsn.1	$⊢ A \in V$
	iunxsn.2	$⊢ x = A \to B = C$
Assertion	iunxsn	$⊢ ⋃_{x \in \{A\}} B = C$

Proof

Step	Hyp	Ref	Expression
1		iunxsn.1	$⊢ A \in V$
2		iunxsn.2	$⊢ x = A \to B = C$
3	2	iunxsng	$⊢ A \in V \to ⋃_{x \in \{A\}} B = C$
4	1 3	ax-mp	$⊢ ⋃_{x \in \{A\}} B = C$