Theorem pwuninel 7401

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 7400. (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		pwexr 6974	. . 3 |- ( ~P U. A e. A -> U. A e. _V )
2		pwuninel2 7400	. . 3 |- ( U. A e. _V -> -. ~P U. A e. A )
3	1, 2	syl 17	. 2 |- ( ~P U. A e. A -> -. ~P U. A e. A )
4		id 22	. 2 |- ( -. ~P U. A e. A -> -. ~P U. A e. A )
5	3, 4	pm2.61i 176	1 |- -. ~P U. A e. A

Syntax hints:   -. wn 3   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  undefnel2  7403  disjen  8117  pnfnre  10081  kelac2lem  37634  kelac2  37635