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Metamath Proof Explorer

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)

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

Proof

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