Theorem sn0topon

Description: The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Assertion	sn0topon	$$⊢\left\{\varnothing \right\}\in \mathrm{TopOn}\left(\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		pw0	$$⊢𝒫\varnothing =\left\{\varnothing \right\}$$
2		0ex	$$⊢\varnothing \in \mathrm{V}$$
3		distopon	$$⊢\varnothing \in \mathrm{V}\to 𝒫\varnothing \in \mathrm{TopOn}\left(\varnothing \right)$$
4	2 3	ax-mp	$$⊢𝒫\varnothing \in \mathrm{TopOn}\left(\varnothing \right)$$
5	1 4	eqeltrri	$$⊢\left\{\varnothing \right\}\in \mathrm{TopOn}\left(\varnothing \right)$$