Theorem sn0cld

Description: The closed sets of the topology { (/) } . (Contributed by FL, 5-Jan-2009)

	Ref	Expression
Assertion	sn0cld	$$⊢\mathrm{Clsd}\left(\left\{\varnothing \right\}\right)=\left\{\varnothing \right\}$$

Proof

Step	Hyp	Ref	Expression
1		0ex	$$⊢\varnothing \in \mathrm{V}$$
2		discld	$$⊢\varnothing \in \mathrm{V}\to \mathrm{Clsd}\left(𝒫\varnothing \right)=𝒫\varnothing$$
3	1 2	ax-mp	$$⊢\mathrm{Clsd}\left(𝒫\varnothing \right)=𝒫\varnothing$$
4		pw0	$$⊢𝒫\varnothing =\left\{\varnothing \right\}$$
5	4	fveq2i	$$⊢\mathrm{Clsd}\left(𝒫\varnothing \right)=\mathrm{Clsd}\left(\left\{\varnothing \right\}\right)$$
6	3 5 4	3eqtr3i	$$⊢\mathrm{Clsd}\left(\left\{\varnothing \right\}\right)=\left\{\varnothing \right\}$$