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Theorem djudisj 5074
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 4790 . 2  |-  U_ x  e.  A  ( {
x }  X.  C
)  C_  ( A  X.  _V )
2 incom 3342 . . 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 4790 . . . 4  |-  U_ y  e.  B  ( {
y }  X.  D
)  C_  ( B  X.  _V )
4 incom 3342 . . . . 5  |-  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )
5 xpdisj1 5071 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  ( B  X.  _V ) )  =  (/) )
64, 5eqtrid 2234 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( B  X.  _V )  i^i  ( A  X.  _V ) )  =  (/) )
7 ssdisj 3494 . . . 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 414 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( U_ y  e.  B  ( { y }  X.  D )  i^i  ( A  X.  _V ) )  =  (/) )
92, 8eqtrid 2234 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  _V )  i^i  U_ y  e.  B  ( { y }  X.  D ) )  =  (/) )
10 ssdisj 3494 . 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 414 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 1364   _Vcvv 2752    i^i cin 3143    C_ wss 3144   (/)c0 3437   {csn 3607   U_ciun 3901    X. cxp 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-iun 3903  df-opab 4080  df-xp 4650  df-rel 4651
This theorem is referenced by: (None)
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