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Theorem difex2 4714
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 4351 . 2  |-  ( A  e.  _V  ->  ( A  \  B )  e. 
_V )
2 ssun2 3511 . . . . 5  |-  A  C_  ( B  u.  A
)
3 uncom 3491 . . . . . 6  |-  ( ( A  \  B )  u.  B )  =  ( B  u.  ( A  \  B ) )
4 undif2 3704 . . . . . 6  |-  ( B  u.  ( A  \  B ) )  =  ( B  u.  A
)
53, 4eqtr2i 2457 . . . . 5  |-  ( B  u.  A )  =  ( ( A  \  B )  u.  B
)
62, 5sseqtri 3380 . . . 4  |-  A  C_  ( ( A  \  B )  u.  B
)
7 unexg 4710 . . . 4  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  ( ( A  \  B )  u.  B
)  e.  _V )
8 ssexg 4349 . . . 4  |-  ( ( A  C_  ( ( A  \  B )  u.  B )  /\  (
( A  \  B
)  u.  B )  e.  _V )  ->  A  e.  _V )
96, 7, 8sylancr 645 . . 3  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  A  e.  _V )
109expcom 425 . 2  |-  ( B  e.  C  ->  (
( A  \  B
)  e.  _V  ->  A  e.  _V ) )
111, 10impbid2 196 1  |-  ( B  e.  C  ->  ( A  e.  _V  <->  ( A  \  B )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   _Vcvv 2956    \ cdif 3317    u. cun 3318    C_ wss 3320
This theorem is referenced by:  elpwun  4756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-pr 3821  df-uni 4016
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