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Theorem elpwun 4758
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 2966 . 2  |-  ( A  e.  ~P ( B  u.  C )  ->  A  e.  _V )
2 elex 2966 . . 3  |-  ( ( A  \  C )  e.  ~P B  -> 
( A  \  C
)  e.  _V )
3 eldifpw.1 . . . 4  |-  C  e. 
_V
4 difex2 4716 . . . 4  |-  ( C  e.  _V  ->  ( A  e.  _V  <->  ( A  \  C )  e.  _V ) )
53, 4ax-mp 8 . . 3  |-  ( A  e.  _V  <->  ( A  \  C )  e.  _V )
62, 5sylibr 205 . 2  |-  ( ( A  \  C )  e.  ~P B  ->  A  e.  _V )
7 elpwg 3808 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  A 
C_  ( B  u.  C ) ) )
8 difexg 4353 . . . . 5  |-  ( A  e.  _V  ->  ( A  \  C )  e. 
_V )
9 elpwg 3808 . . . . 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 3493 . . . . . 6  |-  ( B  u.  C )  =  ( C  u.  B
)
1211sseq2i 3375 . . . . 5  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
13 ssundif 3713 . . . . 5  |-  ( A 
C_  ( C  u.  B )  <->  ( A  \  C )  C_  B
)
1412, 13bitri 242 . . . 4  |-  ( A 
C_  ( B  u.  C )  <->  ( A  \  C )  C_  B
)
1510, 14syl6rbbr 257 . . 3  |-  ( A  e.  _V  ->  ( A  C_  ( B  u.  C )  <->  ( A  \  C )  e.  ~P B ) )
167, 15bitrd 246 . 2  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  ( A  \  C )  e.  ~P B ) )
171, 6, 16pm5.21nii 344 1  |-  ( A  e.  ~P ( B  u.  C )  <->  ( A  \  C )  e.  ~P B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1726   _Vcvv 2958    \ cdif 3319    u. cun 3320    C_ wss 3322   ~Pcpw 3801
This theorem is referenced by:  pwfilem  7403  elrfi  26750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-pr 3823  df-uni 4018
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