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 U. A e. A

Proof

Step	Hyp	Ref	Expression
1		pwexr	|- ( ~P U. A e. A -> U. A e. _V )
2		pwuninel2	|- ( U. A e. _V -> -. ~P U. A e. A )
3	1 2	syl	|- ( ~P U. A e. A -> -. ~P U. A e. A )
4		id	|- ( -. ~P U. A e. A -> -. ~P U. A e. A )
5	3 4	pm2.61i	|- -. ~P U. A e. A