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Theorem difex2 6602
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 4601 . 2  |-  ( A  e.  _V  ->  ( A  \  B )  e. 
_V )
2 ssun2 3673 . . . . 5  |-  A  C_  ( B  u.  A
)
3 uncom 3653 . . . . . 6  |-  ( ( A  \  B )  u.  B )  =  ( B  u.  ( A  \  B ) )
4 undif2 3909 . . . . . 6  |-  ( B  u.  ( A  \  B ) )  =  ( B  u.  A
)
53, 4eqtr2i 2497 . . . . 5  |-  ( B  u.  A )  =  ( ( A  \  B )  u.  B
)
62, 5sseqtri 3541 . . . 4  |-  A  C_  ( ( A  \  B )  u.  B
)
7 unexg 6596 . . . 4  |-  ( ( ( A  \  B
)  e.  _V  /\  B  e.  C )  ->  ( ( A  \  B )  u.  B
)  e.  _V )
8 ssexg 4599 . . . 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 1767   _Vcvv 3118    \ cdif 3478    u. cun 3479    C_ wss 3481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-sn 4034  df-pr 4036  df-uni 4252
This theorem is referenced by:  elpwun  6608
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