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Theorem difex2 6382
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 4439 . 2  |-  ( A  e.  _V  ->  ( A  \  B )  e. 
_V )
2 ssun2 3519 . . . . 5  |-  A  C_  ( B  u.  A
)
3 uncom 3499 . . . . . 6  |-  ( ( A  \  B )  u.  B )  =  ( B  u.  ( A  \  B ) )
4 undif2 3754 . . . . . 6  |-  ( B  u.  ( A  \  B ) )  =  ( B  u.  A
)
53, 4eqtr2i 2463 . . . . 5  |-  ( B  u.  A )  =  ( ( A  \  B )  u.  B
)
62, 5sseqtri 3387 . . . 4  |-  A  C_  ( ( A  \  B )  u.  B
)
7 unexg 6380 . . . 4  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  ( ( A  \  B )  u.  B
)  e.  _V )
8 ssexg 4437 . . . 4  |-  ( ( A  C_  ( ( A  \  B )  u.  B )  /\  (
( A  \  B
)  u.  B )  e.  _V )  ->  A  e.  _V )
96, 7, 8sylancr 663 . . 3  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  A  e.  _V )
109expcom 435 . 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 369    e. wcel 1756   _Vcvv 2971    \ cdif 3324    u. cun 3325    C_ wss 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-sn 3877  df-pr 3879  df-uni 4091
This theorem is referenced by:  elpwun  6388
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