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Theorem elvvuni 4871
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 4869 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2895 . . . . . 6  |-  x  e. 
_V
3 vex 2895 . . . . . 6  |-  y  e. 
_V
42, 3uniop 4393 . . . . 5  |-  U. <. x ,  y >.  =  {
x ,  y }
52, 3opi2 4365 . . . . 5  |-  { x ,  y }  e.  <.
x ,  y >.
64, 5eqeltri 2450 . . . 4  |-  U. <. x ,  y >.  e.  <. x ,  y >.
7 unieq 3959 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  U. A  =  U. <. x ,  y >.
)
8 id 20 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  A  =  <. x ,  y >. )
97, 8eleq12d 2448 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( U. A  e.  A  <->  U. <. x ,  y
>.  e.  <. x ,  y
>. ) )
106, 9mpbiri 225 . . 3  |-  ( A  =  <. x ,  y
>.  ->  U. A  e.  A
)
1110exlimivv 1642 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  U. A  e.  A
)
121, 11sylbi 188 1  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2892   {cpr 3751   <.cop 3753   U.cuni 3950    X. cxp 4809
This theorem is referenced by:  unielxp  6317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rex 2648  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-opab 4201  df-xp 4817
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