MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  djussxp Unicode version

Theorem djussxp 4829
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 3943 . 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 3759 . . 3  |-  ( x  e.  A  ->  { x }  C_  A )
3 ssv 3198 . . 3  |-  B  C_  _V
4 xpss12 4792 . . 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 2613 1  |-  U_ x  e.  A  ( {
x }  X.  B
)  C_  ( A  X.  _V )
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640   U_ciun 3905    X. cxp 4687
This theorem is referenced by:  djudisj  5104  iundom2g  8162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-sn 3646  df-iun 3907  df-opab 4078  df-xp 4695
  Copyright terms: Public domain W3C validator