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Theorem djudisj 4974
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 4692 . 2  |-  U_ x  e.  A  ( {
x }  X.  C
)  C_  ( A  X.  _V )
2 incom 3273 . . 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 4692 . . . 4  |-  U_ y  e.  B  ( {
y }  X.  D
)  C_  ( B  X.  _V )
4 incom 3273 . . . . 5  |-  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )
5 xpdisj1 4971 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )  =  (/) )
64, 5syl5eq 2185 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  (/) )
7 ssdisj 3424 . . . 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 411 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( U_ y  e.  B  ( { y }  X.  D )  i^i  ( A  X.  _V ) )  =  (/) )
92, 8syl5eq 2185 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
10 ssdisj 3424 . 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 411 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 1332   _Vcvv 2689    i^i cin 3075    C_ wss 3076   (/)c0 3368   {csn 3532   U_ciun 3821    X. cxp 4545
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-iun 3823  df-opab 3998  df-xp 4553  df-rel 4554
This theorem is referenced by: (None)
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