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Theorem djussxp 4594
Description: Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
Assertion
Ref Expression
djussxp  |-  U_ x  e.  A  ( {
x }  X.  B
)  C_  ( A  X.  _V )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem djussxp
StepHypRef Expression
1 iunss 3777 . 2  |-  ( U_ x  e.  A  ( { x }  X.  B )  C_  ( A  X.  _V )  <->  A. x  e.  A  ( {
x }  X.  B
)  C_  ( A  X.  _V ) )
2 snssi 3587 . . 3  |-  ( x  e.  A  ->  { x }  C_  A )
3 ssv 3047 . . 3  |-  B  C_  _V
4 xpss12 4558 . . 3  |-  ( ( { x }  C_  A  /\  B  C_  _V )  ->  ( { x }  X.  B )  C_  ( A  X.  _V )
)
52, 3, 4sylancl 405 . 2  |-  ( x  e.  A  ->  ( { x }  X.  B )  C_  ( A  X.  _V ) )
61, 5mprgbir 2434 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  C_  ( A  X.  _V )
Colors of variables: wff set class
Syntax hints:    e. wcel 1439   _Vcvv 2620    C_ wss 3000   {csn 3450   U_ciun 3736    X. cxp 4450
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-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-sn 3456  df-iun 3738  df-opab 3906  df-xp 4458
This theorem is referenced by:  djudisj  4871
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