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Theorem djussxp 4911
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 4024 . 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 3838 . . 3  |-  ( x  e.  A  ->  { x }  C_  A )
3 ssv 3274 . . 3  |-  B  C_  _V
4 xpss12 4874 . . 3  |-  ( ( { x }  C_  A  /\  B  C_  _V )  ->  ( { x }  X.  B )  C_  ( A  X.  _V )
)
52, 3, 4sylancl 643 . 2  |-  ( x  e.  A  ->  ( { x }  X.  B )  C_  ( A  X.  _V ) )
61, 5mprgbir 2689 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  C_  ( A  X.  _V )
Colors of variables: wff set class
Syntax hints:    e. wcel 1710   _Vcvv 2864    C_ wss 3228   {csn 3716   U_ciun 3986    X. cxp 4769
This theorem is referenced by:  djudisj  5186  iundom2g  8252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-v 2866  df-in 3235  df-ss 3242  df-sn 3722  df-iun 3988  df-opab 4159  df-xp 4777
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