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Theorem undifss 3495
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 3253 . . . 4  |-  ( B 
\  A )  C_  B
21jctr 313 . . 3  |-  ( A 
C_  B  ->  ( A  C_  B  /\  ( B  \  A )  C_  B ) )
3 unss 3301 . . 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 3290 . . 3  |-  A  C_  ( A  u.  ( B  \  A ) )
6 sstr 3155 . . 3  |-  ( ( A  C_  ( A  u.  ( B  \  A
) )  /\  ( A  u.  ( B  \  A ) )  C_  B )  ->  A  C_  B )
75, 6mpan 422 . 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 3118    u. cun 3119    C_ wss 3121
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134
This theorem is referenced by:  difsnss  3726  exmidundif  4192  exmidundifim  4193  undifdcss  6900
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