Theorem pp0ex

Description: The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993)

	Ref	Expression
Assertion	pp0ex	$$⊢\left\{\varnothing ,\left\{\varnothing \right\}\right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		pwpw0	$$⊢𝒫\left\{\varnothing \right\}=\left\{\varnothing ,\left\{\varnothing \right\}\right\}$$
2		p0ex	$$⊢\left\{\varnothing \right\}\in \mathrm{V}$$
3	2	pwex	$$⊢𝒫\left\{\varnothing \right\}\in \mathrm{V}$$
4	1 3	eqeltrri	$$⊢\left\{\varnothing ,\left\{\varnothing \right\}\right\}\in \mathrm{V}$$