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Theorem difex2 6506
Description: If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
difex2  |-  ( B  e.  C  ->  ( A  e.  _V  <->  ( A  \  B )  e.  _V ) )

Proof of Theorem difex2
StepHypRef Expression
1 difexg 4513 . 2  |-  ( A  e.  _V  ->  ( A  \  B )  e. 
_V )
2 ssun2 3582 . . . . 5  |-  A  C_  ( B  u.  A
)
3 uncom 3562 . . . . . 6  |-  ( ( A  \  B )  u.  B )  =  ( B  u.  ( A  \  B ) )
4 undif2 3820 . . . . . 6  |-  ( B  u.  ( A  \  B ) )  =  ( B  u.  A
)
53, 4eqtr2i 2412 . . . . 5  |-  ( B  u.  A )  =  ( ( A  \  B )  u.  B
)
62, 5sseqtri 3449 . . . 4  |-  A  C_  ( ( A  \  B )  u.  B
)
7 unexg 6500 . . . 4  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  ( ( A  \  B )  u.  B
)  e.  _V )
8 ssexg 4511 . . . 4  |-  ( ( A  C_  ( ( A  \  B )  u.  B )  /\  (
( A  \  B
)  u.  B )  e.  _V )  ->  A  e.  _V )
96, 7, 8sylancr 661 . . 3  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  A  e.  _V )
109expcom 433 . 2  |-  ( B  e.  C  ->  (
( A  \  B
)  e.  _V  ->  A  e.  _V ) )
111, 10impbid2 204 1  |-  ( B  e.  C  ->  ( A  e.  _V  <->  ( A  \  B )  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1826   _Vcvv 3034    \ cdif 3386    u. cun 3387    C_ wss 3389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-sn 3945  df-pr 3947  df-uni 4164
This theorem is referenced by:  elpwun  6512
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