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Theorem elvvuni 4727
Description: An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
elvvuni  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )

Proof of Theorem elvvuni
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4725 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2766 . . . . . 6  |-  x  e. 
_V
3 vex 2766 . . . . . 6  |-  y  e. 
_V
42, 3uniop 4288 . . . . 5  |-  U. <. x ,  y >.  =  {
x ,  y }
52, 3opi2 4266 . . . . 5  |-  { x ,  y }  e.  <.
x ,  y >.
64, 5eqeltri 2269 . . . 4  |-  U. <. x ,  y >.  e.  <. x ,  y >.
7 unieq 3848 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  U. A  =  U. <. x ,  y >.
)
8 id 19 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  A  =  <. x ,  y >. )
97, 8eleq12d 2267 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( U. A  e.  A  <->  U. <. x ,  y
>.  e.  <. x ,  y
>. ) )
106, 9mpbiri 168 . . 3  |-  ( A  =  <. x ,  y
>.  ->  U. A  e.  A
)
1110exlimivv 1911 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  U. A  e.  A
)
121, 11sylbi 121 1  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364   E.wex 1506    e. wcel 2167   _Vcvv 2763   {cpr 3623   <.cop 3625   U.cuni 3839    X. cxp 4661
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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-opab 4095  df-xp 4669
This theorem is referenced by:  unielxp  6232
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