Theorem elpwun 6977

Description: Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)

Hypothesis

Ref	Expression
eldifpw.1	|- C e. _V

Assertion

Ref	Expression
elpwun	|- ( A e. ~P ( B u. C ) <-> ( A \ C ) e. ~P B )

Proof of Theorem elpwun

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. ~P ( B u. C ) -> A e. _V )
2		elex 3212	. . 3 |- ( ( A \ C ) e. ~P B -> ( A \ C ) e. _V )
3		eldifpw.1	. . . 4 |- C e. _V
4		difex2 6969	. . . 4 |- ( C e. _V -> ( A e. _V <-> ( A \ C ) e. _V ) )
5	3, 4	ax-mp 5	. . 3 |- ( A e. _V <-> ( A \ C ) e. _V )
6	2, 5	sylibr 224	. 2 |- ( ( A \ C ) e. ~P B -> A e. _V )
7		elpwg 4166	. . 3 |- ( A e. _V -> ( A e. ~P ( B u. C ) <-> A C_ ( B u. C ) ) )
8		difexg 4808	. . . . 5 |- ( A e. _V -> ( A \ C ) e. _V )
9		elpwg 4166	. . . . 5 |- ( ( A \ C ) e. _V -> ( ( A \ C ) e. ~P B <-> ( A \ C ) C_ B ) )
10	8, 9	syl 17	. . . 4 |- ( A e. _V -> ( ( A \ C ) e. ~P B <-> ( A \ C ) C_ B ) )
11		uncom 3757	. . . . . 6 |- ( B u. C ) = ( C u. B )
12	11	sseq2i 3630	. . . . 5 |- ( A C_ ( B u. C ) <-> A C_ ( C u. B ) )
13		ssundif 4052	. . . . 5 |- ( A C_ ( C u. B ) <-> ( A \ C ) C_ B )
14	12, 13	bitri 264	. . . 4 |- ( A C_ ( B u. C ) <-> ( A \ C ) C_ B )
15	10, 14	syl6rbbr 279	. . 3 |- ( A e. _V -> ( A C_ ( B u. C ) <-> ( A \ C ) e. ~P B ) )
16	7, 15	bitrd 268	. 2 |- ( A e. _V -> ( A e. ~P ( B u. C ) <-> ( A \ C ) e. ~P B ) )
17	1, 6, 16	pm5.21nii 368	1 |- ( A e. ~P ( B u. C ) <-> ( A \ C ) e. ~P B )

Syntax hints:   <-> wb 196   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   ~Pcpw 4158

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  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:  pwfilem  8260  elrfi  37257  dssmapnvod  38314