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Theorem difex2 4525
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 4162 . 2  |-  ( A  e.  _V  ->  ( A  \  B )  e. 
_V )
2 ssun2 3339 . . . . 5  |-  A  C_  ( B  u.  A
)
3 uncom 3319 . . . . . 6  |-  ( ( A  \  B )  u.  B )  =  ( B  u.  ( A  \  B ) )
4 undif2 3530 . . . . . 6  |-  ( B  u.  ( A  \  B ) )  =  ( B  u.  A
)
53, 4eqtr2i 2304 . . . . 5  |-  ( B  u.  A )  =  ( ( A  \  B )  u.  B
)
62, 5sseqtri 3210 . . . 4  |-  A  C_  ( ( A  \  B )  u.  B
)
7 unexg 4521 . . . 4  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  ( ( A  \  B )  u.  B
)  e.  _V )
8 ssexg 4160 . . . 4  |-  ( ( A  C_  ( ( A  \  B )  u.  B )  /\  (
( A  \  B
)  u.  B )  e.  _V )  ->  A  e.  _V )
96, 7, 8sylancr 644 . . 3  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  A  e.  _V )
109expcom 424 . 2  |-  ( B  e.  C  ->  (
( A  \  B
)  e.  _V  ->  A  e.  _V ) )
111, 10impbid2 195 1  |-  ( B  e.  C  ->  ( A  e.  _V  <->  ( A  \  B )  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152
This theorem is referenced by:  elpwun  4567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828
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