Theorem pwuninel

Description: The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 . (Contributed by NM, 27-Jun-2008) (Proof shortened by Mario Carneiro, 23-Dec-2016)

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

Proof

Step	Hyp	Ref	Expression
1		pwexr	$$⊢𝒫\bigcup A\in A\to \bigcup A\in \mathrm{V}$$
2		pwuninel2	$$⊢\bigcup A\in \mathrm{V}\to \neg 𝒫\bigcup A\in A$$
3	1 2	syl	$$⊢𝒫\bigcup A\in A\to \neg 𝒫\bigcup A\in A$$
4		id	$$⊢\neg 𝒫\bigcup A\in A\to \neg 𝒫\bigcup A\in A$$
5	3 4	pm2.61i	$$⊢\neg 𝒫\bigcup A\in A$$