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Theorem disjssun 3687
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 3589 . . . . 5  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
21equncomi 3495 . . . 4  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  C )  u.  ( A  i^i  B ) )
3 uneq2 3497 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( ( A  i^i  C
)  u.  (/) ) )
4 un0 3654 . . . . 5  |-  ( ( A  i^i  C )  u.  (/) )  =  ( A  i^i  C )
53, 4syl6eq 2486 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( A  i^i  C ) )
62, 5syl5eq 2482 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( A  i^i  ( B  u.  C ) )  =  ( A  i^i  C
) )
76eqeq1d 2446 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  ( B  u.  C ) )  =  A  <->  ( A  i^i  C )  =  A ) )
8 df-ss 3336 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  i^i  ( B  u.  C
) )  =  A )
9 df-ss 3336 . 2  |-  ( A 
C_  C  <->  ( A  i^i  C )  =  A )
107, 8, 93bitr4g 281 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
C_  ( B  u.  C )  <->  A  C_  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630
This theorem is referenced by:  hashbclem  11703  alexsubALTlem2  18081  iccntr  18854  reconnlem1  18859  dvne0  19897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631
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