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Theorem ssundifim 3347
Description: A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
ssundifim  |-  ( A 
C_  ( B  u.  C )  ->  ( A  \  B )  C_  C )

Proof of Theorem ssundifim
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pm5.6r 870 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  B  \/  x  e.  C
) )  ->  (
( x  e.  A  /\  -.  x  e.  B
)  ->  x  e.  C ) )
2 elun 3125 . . . . 5  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
32imbi2i 224 . . . 4  |-  ( ( x  e.  A  ->  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  ->  ( x  e.  B  \/  x  e.  C ) ) )
4 eldif 2993 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
54imbi1i 236 . . . 4  |-  ( ( x  e.  ( A 
\  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  x  e.  C ) )
61, 3, 53imtr4i 199 . . 3  |-  ( ( x  e.  A  ->  x  e.  ( B  u.  C ) )  -> 
( x  e.  ( A  \  B )  ->  x  e.  C
) )
76alimi 1385 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  ( B  u.  C
) )  ->  A. x
( x  e.  ( A  \  B )  ->  x  e.  C
) )
8 dfss2 2999 . 2  |-  ( A 
C_  ( B  u.  C )  <->  A. x
( x  e.  A  ->  x  e.  ( B  u.  C ) ) )
9 dfss2 2999 . 2  |-  ( ( A  \  B ) 
C_  C  <->  A. x
( x  e.  ( A  \  B )  ->  x  e.  C
) )
107, 8, 93imtr4i 199 1  |-  ( A 
C_  ( B  u.  C )  ->  ( A  \  B )  C_  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662   A.wal 1283    e. wcel 1434    \ cdif 2981    u. cun 2982    C_ wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997
This theorem is referenced by: (None)
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