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Theorem elvvuni 4766
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 4764 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2804 . . . . . 6  |-  x  e. 
_V
3 vex 2804 . . . . . 6  |-  y  e. 
_V
42, 3uniop 4285 . . . . 5  |-  U. <. x ,  y >.  =  {
x ,  y }
52, 3opi2 4257 . . . . 5  |-  { x ,  y }  e.  <.
x ,  y >.
64, 5eqeltri 2366 . . . 4  |-  U. <. x ,  y >.  e.  <. x ,  y >.
7 unieq 3852 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  U. A  =  U. <. x ,  y >.
)
8 id 19 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  A  =  <. x ,  y >. )
97, 8eleq12d 2364 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( U. A  e.  A  <->  U. <. x ,  y
>.  e.  <. x ,  y
>. ) )
106, 9mpbiri 224 . . 3  |-  ( A  =  <. x ,  y
>.  ->  U. A  e.  A
)
1110exlimivv 1625 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  U. A  e.  A
)
121, 11sylbi 187 1  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   {cpr 3654   <.cop 3656   U.cuni 3843    X. cxp 4703
This theorem is referenced by:  unielxp  6174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-opab 4094  df-xp 4711
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