Theorem pwuninel

Description: The powerclass of the union of a class does not belong to that class. This theorem provides a way of constructing a new set that does not belong to a given set. See also pwuninel2 . (Contributed by NM, 27-Jun-2008) (Proof shortened by Mario Carneiro, 23-Dec-2016) Avoid ax-pr and ax-un . (Revised by Umit Teoman Dogan, 10-Jun-2026)

	Ref	Expression
Assertion	pwuninel	$$⊢\neg 𝒫\bigcup A\in A$$

Proof

Step	Hyp	Ref	Expression
1		elssuni	$$⊢𝒫\bigcup A\in A\to 𝒫\bigcup A\subseteq \bigcup A$$
2	1	sspwd	$$⊢𝒫\bigcup A\in A\to 𝒫𝒫\bigcup A\subseteq 𝒫\bigcup A$$
3		pwnss	$$⊢𝒫\bigcup A\in A\to \neg 𝒫𝒫\bigcup A\subseteq 𝒫\bigcup A$$
4	2 3	pm2.65i	$$⊢\neg 𝒫\bigcup A\in A$$