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Theorem djudisj 4810
Description: Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)
Assertion
Ref Expression
djudisj  |-  ( ( A  i^i  B )  =  (/)  ->  ( U_ x  e.  A  ( { x }  X.  C )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
Distinct variable groups:    x, A    y, B
Allowed substitution hints:    A( y)    B( x)    C( x, y)    D( x, y)

Proof of Theorem djudisj
StepHypRef Expression
1 djussxp 4537 . 2  |-  U_ x  e.  A  ( {
x }  X.  C
)  C_  ( A  X.  _V )
2 incom 3176 . . 3  |-  ( ( A  X.  _V )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  ( U_ y  e.  B  ( { y }  X.  D )  i^i  ( A  X.  _V ) )
3 djussxp 4537 . . . 4  |-  U_ y  e.  B  ( {
y }  X.  D
)  C_  ( B  X.  _V )
4 incom 3176 . . . . 5  |-  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )
5 xpdisj1 4807 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )  =  (/) )
64, 5syl5eq 2127 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  (/) )
7 ssdisj 3321 . . . 4  |-  ( (
U_ y  e.  B  ( { y }  X.  D )  C_  ( B  X.  _V )  /\  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  (/) )  ->  ( U_ y  e.  B  ( { y }  X.  D )  i^i  ( A  X.  _V ) )  =  (/) )
83, 6, 7sylancr 405 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( U_ y  e.  B  ( { y }  X.  D )  i^i  ( A  X.  _V ) )  =  (/) )
92, 8syl5eq 2127 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
10 ssdisj 3321 . 2  |-  ( (
U_ x  e.  A  ( { x }  X.  C )  C_  ( A  X.  _V )  /\  ( ( A  X.  _V )  i^i  U_ y  e.  B  ( {
y }  X.  D
) )  =  (/) )  ->  ( U_ x  e.  A  ( {
x }  X.  C
)  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
111, 9, 10sylancr 405 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( U_ x  e.  A  ( { x }  X.  C )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   _Vcvv 2612    i^i cin 2983    C_ wss 2984   (/)c0 3269   {csn 3422   U_ciun 3704    X. cxp 4397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-iun 3706  df-opab 3866  df-xp 4405  df-rel 4406
This theorem is referenced by: (None)
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