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Theorem djudisj 4902
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 4622 . 2  |-  U_ x  e.  A  ( {
x }  X.  C
)  C_  ( A  X.  _V )
2 incom 3215 . . 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 4622 . . . 4  |-  U_ y  e.  B  ( {
y }  X.  D
)  C_  ( B  X.  _V )
4 incom 3215 . . . . 5  |-  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )
5 xpdisj1 4899 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )  =  (/) )
64, 5syl5eq 2144 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  (/) )
7 ssdisj 3366 . . . 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 408 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( U_ y  e.  B  ( { y }  X.  D )  i^i  ( A  X.  _V ) )  =  (/) )
92, 8syl5eq 2144 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
10 ssdisj 3366 . 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 408 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 1299   _Vcvv 2641    i^i cin 3020    C_ wss 3021   (/)c0 3310   {csn 3474   U_ciun 3760    X. cxp 4475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-iun 3762  df-opab 3930  df-xp 4483  df-rel 4484
This theorem is referenced by: (None)
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