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Theorem undifss 3411
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 3170 . . . 4  |-  ( B 
\  A )  C_  B
21jctr 311 . . 3  |-  ( A 
C_  B  ->  ( A  C_  B  /\  ( B  \  A )  C_  B ) )
3 unss 3218 . . 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 3207 . . 3  |-  A  C_  ( A  u.  ( B  \  A ) )
6 sstr 3073 . . 3  |-  ( ( A  C_  ( A  u.  ( B  \  A
) )  /\  ( A  u.  ( B  \  A ) )  C_  B )  ->  A  C_  B )
75, 6mpan 418 . 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 3036    u. cun 3037    C_ wss 3039
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052
This theorem is referenced by:  difsnss  3634  exmidundif  4097  exmidundifim  4098  undifdcss  6777
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