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Theorem undifss 3474
Description: Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undifss  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  C_  B
)

Proof of Theorem undifss
StepHypRef Expression
1 difss 3233 . . . 4  |-  ( B 
\  A )  C_  B
21jctr 313 . . 3  |-  ( A 
C_  B  ->  ( A  C_  B  /\  ( B  \  A )  C_  B ) )
3 unss 3281 . . 3  |-  ( ( A  C_  B  /\  ( B  \  A ) 
C_  B )  <->  ( A  u.  ( B  \  A
) )  C_  B
)
42, 3sylib 121 . 2  |-  ( A 
C_  B  ->  ( A  u.  ( B  \  A ) )  C_  B )
5 ssun1 3270 . . 3  |-  A  C_  ( A  u.  ( B  \  A ) )
6 sstr 3136 . . 3  |-  ( ( A  C_  ( A  u.  ( B  \  A
) )  /\  ( A  u.  ( B  \  A ) )  C_  B )  ->  A  C_  B )
75, 6mpan 421 . 2  |-  ( ( A  u.  ( B 
\  A ) ) 
C_  B  ->  A  C_  B )
84, 7impbii 125 1  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  C_  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \ cdif 3099    u. cun 3100    C_ wss 3102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115
This theorem is referenced by:  difsnss  3702  exmidundif  4167  exmidundifim  4168  undifdcss  6867
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