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Theorem elpwun 6500
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
StepHypRef Expression
1 elex 3087 . 2  |-  ( A  e.  ~P ( B  u.  C )  ->  A  e.  _V )
2 elex 3087 . . 3  |-  ( ( A  \  C )  e.  ~P B  -> 
( A  \  C
)  e.  _V )
3 eldifpw.1 . . . 4  |-  C  e. 
_V
4 difex2 6494 . . . 4  |-  ( C  e.  _V  ->  ( A  e.  _V  <->  ( A  \  C )  e.  _V ) )
53, 4ax-mp 5 . . 3  |-  ( A  e.  _V  <->  ( A  \  C )  e.  _V )
62, 5sylibr 212 . 2  |-  ( ( A  \  C )  e.  ~P B  ->  A  e.  _V )
7 elpwg 3977 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  A 
C_  ( B  u.  C ) ) )
8 difexg 4549 . . . . 5  |-  ( A  e.  _V  ->  ( A  \  C )  e. 
_V )
9 elpwg 3977 . . . . 5  |-  ( ( A  \  C )  e.  _V  ->  (
( A  \  C
)  e.  ~P B  <->  ( A  \  C ) 
C_  B ) )
108, 9syl 16 . . . 4  |-  ( A  e.  _V  ->  (
( A  \  C
)  e.  ~P B  <->  ( A  \  C ) 
C_  B ) )
11 uncom 3609 . . . . . 6  |-  ( B  u.  C )  =  ( C  u.  B
)
1211sseq2i 3490 . . . . 5  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
13 ssundif 3871 . . . . 5  |-  ( A 
C_  ( C  u.  B )  <->  ( A  \  C )  C_  B
)
1412, 13bitri 249 . . . 4  |-  ( A 
C_  ( B  u.  C )  <->  ( A  \  C )  C_  B
)
1510, 14syl6rbbr 264 . . 3  |-  ( A  e.  _V  ->  ( A  C_  ( B  u.  C )  <->  ( A  \  C )  e.  ~P B ) )
167, 15bitrd 253 . 2  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  ( A  \  C )  e.  ~P B ) )
171, 6, 16pm5.21nii 353 1  |-  ( A  e.  ~P ( B  u.  C )  <->  ( A  \  C )  e.  ~P B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1758   _Vcvv 3078    \ cdif 3434    u. cun 3435    C_ wss 3437   ~Pcpw 3969
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-pw 3971  df-sn 3987  df-pr 3989  df-uni 4201
This theorem is referenced by:  pwfilem  7717  elrfi  29179
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