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	$$⊢A\in \mathrm{V}$$
	iunxsn.2	$$⊢x=A\to B=C$$
Assertion	iunxsn	$$⊢\bigcup _{x\in \left\{A\right\}}B=C$$

Proof

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