Theorem pwuninel 7009

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 7008. (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 3070	. . . 4 |- ( ~P U. A e. A -> ~P U. A e. _V )
2		pwexb 6595	. . . 4 |- ( U. A e. _V <-> ~P U. A e. _V )
3	1, 2	sylibr 214	. . 3 |- ( ~P U. A e. A -> U. A e. _V )
4		pwuninel2 7008	. . 3 |- ( U. A e. _V -> -. ~P U. A e. A )
5	3, 4	syl 17	. 2 |- ( ~P U. A e. A -> -. ~P U. A e. A )
6		id 23	. 2 |- ( -. ~P U. A e. A -> -. ~P U. A e. A )
7	5, 6	pm2.61i 166	1 |- -. ~P U. A e. A

Syntax hints:   -. wn 3   e. wcel 1844   _Vcvv 3061   ~Pcpw 3957   U.cuni 4193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-pw 3959  df-sn 3975  df-pr 3977  df-uni 4194

This theorem is referenced by:  undefnel2  7011  disjen  7714  pnfnre  9667  kelac2lem  35385  kelac2  35386