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Theorem disjssun 3285
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 3184 . . . . 5  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
21equncomi 3089 . . . 4  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  C )  u.  ( A  i^i  B ) )
3 uneq2 3091 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( ( A  i^i  C
)  u.  (/) ) )
4 un0 3251 . . . . 5  |-  ( ( A  i^i  C )  u.  (/) )  =  ( A  i^i  C )
53, 4syl6eq 2088 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( A  i^i  C ) )
62, 5syl5eq 2084 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( A  i^i  ( B  u.  C ) )  =  ( A  i^i  C
) )
76eqeq1d 2048 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  ( B  u.  C ) )  =  A  <->  ( A  i^i  C )  =  A ) )
8 df-ss 2931 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  i^i  ( B  u.  C
) )  =  A )
9 df-ss 2931 . 2  |-  ( A 
C_  C  <->  ( A  i^i  C )  =  A )
107, 8, 93bitr4g 212 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
C_  ( B  u.  C )  <->  A  C_  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    u. cun 2915    i^i cin 2916    C_ wss 2917   (/)c0 3224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225
This theorem is referenced by: (None)
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