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Theorem disjssun 3396
Description: Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
disjssun  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
C_  ( B  u.  C )  <->  A  C_  C
) )

Proof of Theorem disjssun
StepHypRef Expression
1 indi 3293 . . . . 5  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
21equncomi 3192 . . . 4  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  C )  u.  ( A  i^i  B ) )
3 uneq2 3194 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( ( A  i^i  C
)  u.  (/) ) )
4 un0 3366 . . . . 5  |-  ( ( A  i^i  C )  u.  (/) )  =  ( A  i^i  C )
53, 4syl6eq 2166 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( A  i^i  C ) )
62, 5syl5eq 2162 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( A  i^i  ( B  u.  C ) )  =  ( A  i^i  C
) )
76eqeq1d 2126 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  ( B  u.  C ) )  =  A  <->  ( A  i^i  C )  =  A ) )
8 df-ss 3054 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  i^i  ( B  u.  C
) )  =  A )
9 df-ss 3054 . 2  |-  ( A 
C_  C  <->  ( A  i^i  C )  =  A )
107, 8, 93bitr4g 222 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
C_  ( B  u.  C )  <->  A  C_  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    u. cun 3039    i^i cin 3040    C_ wss 3041   (/)c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by: (None)
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