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Theorem undifss 3503
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 3261 . . . 4  |-  ( B 
\  A )  C_  B
21jctr 315 . . 3  |-  ( A 
C_  B  ->  ( A  C_  B  /\  ( B  \  A )  C_  B ) )
3 unss 3309 . . 3  |-  ( ( A  C_  B  /\  ( B  \  A ) 
C_  B )  <->  ( A  u.  ( B  \  A
) )  C_  B
)
42, 3sylib 122 . 2  |-  ( A 
C_  B  ->  ( A  u.  ( B  \  A ) )  C_  B )
5 ssun1 3298 . . 3  |-  A  C_  ( A  u.  ( B  \  A ) )
6 sstr 3163 . . 3  |-  ( ( A  C_  ( A  u.  ( B  \  A
) )  /\  ( A  u.  ( B  \  A ) )  C_  B )  ->  A  C_  B )
75, 6mpan 424 . 2  |-  ( ( A  u.  ( B 
\  A ) ) 
C_  B  ->  A  C_  B )
84, 7impbii 126 1  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  C_  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    \ cdif 3126    u. cun 3127    C_ wss 3129
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142
This theorem is referenced by:  difsnss  3738  exmidundif  4206  exmidundifim  4207  undifdcss  6921
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