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Theorem djussxp 4767
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 3925 . 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 3735 . . 3  |-  ( x  e.  A  ->  { x }  C_  A )
3 ssv 3177 . . 3  |-  B  C_  _V
4 xpss12 4729 . . 3  |-  ( ( { x }  C_  A  /\  B  C_  _V )  ->  ( { x }  X.  B )  C_  ( A  X.  _V )
)
52, 3, 4sylancl 413 . 2  |-  ( x  e.  A  ->  ( { x }  X.  B )  C_  ( A  X.  _V ) )
61, 5mprgbir 2535 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  C_  ( A  X.  _V )
Colors of variables: wff set class
Syntax hints:    e. wcel 2148   _Vcvv 2737    C_ wss 3129   {csn 3591   U_ciun 3884    X. cxp 4620
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-sn 3597  df-iun 3886  df-opab 4062  df-xp 4628
This theorem is referenced by:  djudisj  5051
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