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Theorem djudisj 5238
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 4959 . 2  |-  U_ x  e.  A  ( {
x }  X.  C
)  C_  ( A  X.  _V )
2 incom 3477 . . 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 4959 . . . 4  |-  U_ y  e.  B  ( {
y }  X.  D
)  C_  ( B  X.  _V )
4 incom 3477 . . . . 5  |-  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )
5 xpdisj1 5235 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )  =  (/) )
64, 5syl5eq 2432 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  (/) )
7 ssdisj 3621 . . . 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 645 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( U_ y  e.  B  ( { y }  X.  D )  i^i  ( A  X.  _V ) )  =  (/) )
92, 8syl5eq 2432 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
10 ssdisj 3621 . 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 645 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 1649   _Vcvv 2900    i^i cin 3263    C_ wss 3264   (/)c0 3572   {csn 3758   U_ciun 4036    X. cxp 4817
This theorem is referenced by:  ackbij1lem9  8042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-iun 4038  df-opab 4209  df-xp 4825  df-rel 4826
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