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Theorem elvvuni 5000
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 4998 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 3074 . . . . . 6  |-  x  e. 
_V
3 vex 3074 . . . . . 6  |-  y  e. 
_V
42, 3uniop 4695 . . . . 5  |-  U. <. x ,  y >.  =  {
x ,  y }
52, 3opi2 4661 . . . . 5  |-  { x ,  y }  e.  <.
x ,  y >.
64, 5eqeltri 2535 . . . 4  |-  U. <. x ,  y >.  e.  <. x ,  y >.
7 unieq 4200 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  U. A  =  U. <. x ,  y >.
)
8 id 22 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  A  =  <. x ,  y >. )
97, 8eleq12d 2533 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( U. A  e.  A  <->  U. <. x ,  y
>.  e.  <. x ,  y
>. ) )
106, 9mpbiri 233 . . 3  |-  ( A  =  <. x ,  y
>.  ->  U. A  e.  A
)
1110exlimivv 1690 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  U. A  e.  A
)
121, 11sylbi 195 1  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3071   {cpr 3980   <.cop 3984   U.cuni 4192    X. cxp 4939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rex 2801  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-opab 4452  df-xp 4947
This theorem is referenced by:  unielxp  6715
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