Theorem pwuninel 7001

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 7000. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
pwuninel	|- -. ~P U. A e. A

Proof of Theorem pwuninel

Step	Hyp	Ref	Expression
1		elex 3122	. . . 4 |- ( ~P U. A e. A -> ~P U. A e. _V )
2		pwexb 6589	. . . 4 |- ( U. A e. _V <-> ~P U. A e. _V )
3	1, 2	sylibr 212	. . 3 |- ( ~P U. A e. A -> U. A e. _V )
4		pwuninel2 7000	. . 3 |- ( U. A e. _V -> -. ~P U. A e. A )
5	3, 4	syl 16	. 2 |- ( ~P U. A e. A -> -. ~P U. A e. A )
6		id 22	. 2 |- ( -. ~P U. A e. A -> -. ~P U. A e. A )
7	5, 6	pm2.61i 164	1 |- -. ~P U. A e. A

Syntax hints:   -. wn 3   e. wcel 1767   _Vcvv 3113   ~Pcpw 4010   U.cuni 4245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030  df-uni 4246

This theorem is referenced by:  undefnel2  7003  disjen  7671  pnfnre  9631  kelac2lem  30614  kelac2  30615