Theorem grusn

Description: A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013)

	Ref	Expression
Assertion	grusn	$$⊢\left(U\in \mathrm{Univ}\wedge A\in U\right)\to \left\{A\right\}\in U$$

Proof

Step	Hyp	Ref	Expression
1		dfsn2	$$⊢\left\{A\right\}=\left\{A,A\right\}$$
2		grupr	$$⊢\left(U\in \mathrm{Univ}\wedge A\in U\wedge A\in U\right)\to \left\{A,A\right\}\in U$$
3	2	3anidm23	$$⊢\left(U\in \mathrm{Univ}\wedge A\in U\right)\to \left\{A,A\right\}\in U$$
4	1 3	eqeltrid	$$⊢\left(U\in \mathrm{Univ}\wedge A\in U\right)\to \left\{A\right\}\in U$$