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Theorem disjssun 3421
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 3318 . . . . 5  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
21equncomi 3217 . . . 4  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  C )  u.  ( A  i^i  B ) )
3 uneq2 3219 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( ( A  i^i  C
)  u.  (/) ) )
4 un0 3391 . . . . 5  |-  ( ( A  i^i  C )  u.  (/) )  =  ( A  i^i  C )
53, 4syl6eq 2186 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( A  i^i  C ) )
62, 5syl5eq 2182 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( A  i^i  ( B  u.  C ) )  =  ( A  i^i  C
) )
76eqeq1d 2146 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  ( B  u.  C ) )  =  A  <->  ( A  i^i  C )  =  A ) )
8 df-ss 3079 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  i^i  ( B  u.  C
) )  =  A )
9 df-ss 3079 . 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 1331    u. cun 3064    i^i cin 3065    C_ wss 3066   (/)c0 3358
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359
This theorem is referenced by: (None)
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