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	⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

Proof

Step	Hyp	Ref	Expression
1		elssuni	⊢ ( 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 )
2	1	sspwd	⊢ ( 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ 𝒫 ∪ 𝐴 )
3		pwnss	⊢ ( 𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 𝒫 ∪ 𝐴 ⊆ 𝒫 ∪ 𝐴 )
4	2 3	pm2.65i	⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴