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	⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

Proof

Step	Hyp	Ref	Expression
1		pwexr	⊢ ( 𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V )
2		pwuninel2	⊢ ( ∪ 𝐴 ∈ V → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 )
3	1 2	syl	⊢ ( 𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 )
4		id	⊢ ( ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 )
5	3 4	pm2.61i	⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴