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Theorem elpwun 6598
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 3104 . 2  |-  ( A  e.  ~P ( B  u.  C )  ->  A  e.  _V )
2 elex 3104 . . 3  |-  ( ( A  \  C )  e.  ~P B  -> 
( A  \  C
)  e.  _V )
3 eldifpw.1 . . . 4  |-  C  e. 
_V
4 difex2 6592 . . . 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 4005 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  A 
C_  ( B  u.  C ) ) )
8 difexg 4585 . . . . 5  |-  ( A  e.  _V  ->  ( A  \  C )  e. 
_V )
9 elpwg 4005 . . . . 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 3633 . . . . . 6  |-  ( B  u.  C )  =  ( C  u.  B
)
1211sseq2i 3514 . . . . 5  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
13 ssundif 3897 . . . . 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 1804   _Vcvv 3095    \ cdif 3458    u. cun 3459    C_ wss 3461   ~Pcpw 3997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-pw 3999  df-sn 4015  df-pr 4017  df-uni 4235
This theorem is referenced by:  pwfilem  7816  elrfi  30601
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