Theorem unisng

Description: A set equals the union of its singleton. Theorem 8.2 of Quine p. 53. (Contributed by NM, 13-Aug-2002)

	Ref	Expression
Assertion	unisng	$$⊢A\in V\to \bigcup \left\{A\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		dfsn2	$$⊢\left\{A\right\}=\left\{A,A\right\}$$
2	1	unieqi	$$⊢\bigcup \left\{A\right\}=\bigcup \left\{A,A\right\}$$
3	2	a1i	$$⊢A\in V\to \bigcup \left\{A\right\}=\bigcup \left\{A,A\right\}$$
4		uniprg	$$⊢\left(A\in V\wedge A\in V\right)\to \bigcup \left\{A,A\right\}=A\cup A$$
5	4	anidms	$$⊢A\in V\to \bigcup \left\{A,A\right\}=A\cup A$$
6		unidm	$$⊢A\cup A=A$$
7	6	a1i	$$⊢A\in V\to A\cup A=A$$
8	3 5 7	3eqtrd	$$⊢A\in V\to \bigcup \left\{A\right\}=A$$