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 U. A e. A

Proof

Step	Hyp	Ref	Expression
1		elssuni	|- ( ~P U. A e. A -> ~P U. A C_ U. A )
2	1	sspwd	|- ( ~P U. A e. A -> ~P ~P U. A C_ ~P U. A )
3		pwnss	|- ( ~P U. A e. A -> -. ~P ~P U. A C_ ~P U. A )
4	2 3	pm2.65i	|- -. ~P U. A e. A