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Theorem elvvuni 5049
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 5047 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 3109 . . . . . 6  |-  x  e. 
_V
3 vex 3109 . . . . . 6  |-  y  e. 
_V
42, 3uniop 4739 . . . . 5  |-  U. <. x ,  y >.  =  {
x ,  y }
52, 3opi2 4705 . . . . 5  |-  { x ,  y }  e.  <.
x ,  y >.
64, 5eqeltri 2538 . . . 4  |-  U. <. x ,  y >.  e.  <. x ,  y >.
7 unieq 4243 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  U. A  =  U. <. x ,  y >.
)
8 id 22 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  A  =  <. x ,  y >. )
97, 8eleq12d 2536 . . . 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 1728 . 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 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106   {cpr 4018   <.cop 4022   U.cuni 4235    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-opab 4498  df-xp 4994
This theorem is referenced by:  unielxp  6809
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