MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elpwun Unicode version

Theorem elpwun 4583
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 2809 . 2  |-  ( A  e.  ~P ( B  u.  C )  ->  A  e.  _V )
2 elex 2809 . . 3  |-  ( ( A  \  C )  e.  ~P B  -> 
( A  \  C
)  e.  _V )
3 eldifpw.1 . . . 4  |-  C  e. 
_V
4 difex2 4541 . . . 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 203 . 2  |-  ( ( A  \  C )  e.  ~P B  ->  A  e.  _V )
7 elpwg 3645 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  A 
C_  ( B  u.  C ) ) )
8 difexg 4178 . . . . 5  |-  ( A  e.  _V  ->  ( A  \  C )  e. 
_V )
9 elpwg 3645 . . . . 5  |-  ( ( A  \  C )  e.  _V  ->  (
( A  \  C
)  e.  ~P B  <->  ( A  \  C ) 
C_  B ) )
108, 9syl 15 . . . 4  |-  ( A  e.  _V  ->  (
( A  \  C
)  e.  ~P B  <->  ( A  \  C ) 
C_  B ) )
11 uncom 3332 . . . . . 6  |-  ( B  u.  C )  =  ( C  u.  B
)
1211sseq2i 3216 . . . . 5  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
13 ssundif 3550 . . . . 5  |-  ( A 
C_  ( C  u.  B )  <->  ( A  \  C )  C_  B
)
1412, 13bitri 240 . . . 4  |-  ( A 
C_  ( B  u.  C )  <->  ( A  \  C )  C_  B
)
1510, 14syl6rbbr 255 . . 3  |-  ( A  e.  _V  ->  ( A  C_  ( B  u.  C )  <->  ( A  \  C )  e.  ~P B ) )
167, 15bitrd 244 . 2  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  ( A  \  C )  e.  ~P B ) )
171, 6, 16pm5.21nii 342 1  |-  ( A  e.  ~P ( B  u.  C )  <->  ( A  \  C )  e.  ~P B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   ~Pcpw 3638
This theorem is referenced by:  pwfilem  7166  elrfi  26872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660  df-uni 3844
  Copyright terms: Public domain W3C validator