Theorem pwuninel 6894

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 6893. (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 3077	. . . 4 |- ( ~P U. A e. A -> ~P U. A e. _V )
2		pwexb 6487	. . . 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 6893	. . 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 1758   _Vcvv 3068   ~Pcpw 3958   U.cuni 4189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-pw 3960  df-sn 3976  df-pr 3978  df-uni 4190

This theorem is referenced by:  undefnel2  6896  disjen  7568  pnfnre  9526  kelac2lem  29555  kelac2  29556