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Theorem ssundifim 3518
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 928 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  B  \/  x  e.  C
) )  ->  (
( x  e.  A  /\  -.  x  e.  B
)  ->  x  e.  C ) )
2 elun 3288 . . . . 5  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
32imbi2i 226 . . . 4  |-  ( ( x  e.  A  ->  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  ->  ( x  e.  B  \/  x  e.  C ) ) )
4 eldif 3150 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
54imbi1i 238 . . . 4  |-  ( ( x  e.  ( A 
\  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  x  e.  C ) )
61, 3, 53imtr4i 201 . . 3  |-  ( ( x  e.  A  ->  x  e.  ( B  u.  C ) )  -> 
( x  e.  ( A  \  B )  ->  x  e.  C
) )
76alimi 1465 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  ( B  u.  C
) )  ->  A. x
( x  e.  ( A  \  B )  ->  x  e.  C
) )
8 dfss2 3156 . 2  |-  ( A 
C_  ( B  u.  C )  <->  A. x
( x  e.  A  ->  x  e.  ( B  u.  C ) ) )
9 dfss2 3156 . 2  |-  ( ( A  \  B ) 
C_  C  <->  A. x
( x  e.  ( A  \  B )  ->  x  e.  C
) )
107, 8, 93imtr4i 201 1  |-  ( A 
C_  ( B  u.  C )  ->  ( A  \  B )  C_  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709   A.wal 1361    e. wcel 2158    \ cdif 3138    u. cun 3139    C_ wss 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154
This theorem is referenced by: (None)
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