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Theorem djussxp 4980
Description: Disjoint union is a subset of a Cartesian 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 4206 . 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 4012 . . 3  |-  ( x  e.  A  ->  { x }  C_  A )
3 ssv 3371 . . 3  |-  B  C_  _V
4 xpss12 4940 . . 3  |-  ( ( { x }  C_  A  /\  B  C_  _V )  ->  ( { x }  X.  B )  C_  ( A  X.  _V )
)
52, 3, 4sylancl 662 . 2  |-  ( x  e.  A  ->  ( { x }  X.  B )  C_  ( A  X.  _V ) )
61, 5mprgbir 2781 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  C_  ( A  X.  _V )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1756   _Vcvv 2967    C_ wss 3323   {csn 3872   U_ciun 4166    X. cxp 4833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-v 2969  df-in 3330  df-ss 3337  df-sn 3873  df-iun 4168  df-opab 4346  df-xp 4841
This theorem is referenced by:  djudisj  5260  iundom2g  8696
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