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

Theorem elpwun 6615
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 3090 . 2  |-  ( A  e.  ~P ( B  u.  C )  ->  A  e.  _V )
2 elex 3090 . . 3  |-  ( ( A  \  C )  e.  ~P B  -> 
( A  \  C
)  e.  _V )
3 eldifpw.1 . . . 4  |-  C  e. 
_V
4 difex2 6609 . . . 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 215 . 2  |-  ( ( A  \  C )  e.  ~P B  ->  A  e.  _V )
7 elpwg 3987 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  A 
C_  ( B  u.  C ) ) )
8 difexg 4569 . . . . 5  |-  ( A  e.  _V  ->  ( A  \  C )  e. 
_V )
9 elpwg 3987 . . . . 5  |-  ( ( A  \  C )  e.  _V  ->  (
( A  \  C
)  e.  ~P B  <->  ( A  \  C ) 
C_  B ) )
108, 9syl 17 . . . 4  |-  ( A  e.  _V  ->  (
( A  \  C
)  e.  ~P B  <->  ( A  \  C ) 
C_  B ) )
11 uncom 3610 . . . . . 6  |-  ( B  u.  C )  =  ( C  u.  B
)
1211sseq2i 3489 . . . . 5  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
13 ssundif 3879 . . . . 5  |-  ( A 
C_  ( C  u.  B )  <->  ( A  \  C )  C_  B
)
1412, 13bitri 252 . . . 4  |-  ( A 
C_  ( B  u.  C )  <->  ( A  \  C )  C_  B
)
1510, 14syl6rbbr 267 . . 3  |-  ( A  e.  _V  ->  ( A  C_  ( B  u.  C )  <->  ( A  \  C )  e.  ~P B ) )
167, 15bitrd 256 . 2  |-  ( A  e.  _V  ->  ( A  e.  ~P ( B  u.  C )  <->  ( A  \  C )  e.  ~P B ) )
171, 6, 16pm5.21nii 354 1  |-  ( A  e.  ~P ( B  u.  C )  <->  ( A  \  C )  e.  ~P B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    e. wcel 1868   _Vcvv 3081    \ cdif 3433    u. cun 3434    C_ wss 3436   ~Pcpw 3979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-pw 3981  df-sn 3997  df-pr 3999  df-uni 4217
This theorem is referenced by:  pwfilem  7871  elrfi  35455
  Copyright terms: Public domain W3C validator