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Theorem zfpair2 4630
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4629. See zfpair 4627 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
Assertion
Ref Expression
zfpair2  |-  { x ,  y }  e.  _V

Proof of Theorem zfpair2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-pr 4629 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
21bm1.3ii 4514 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
3 dfcleq 2444 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 3071 . . . . . . . 8  |-  w  e. 
_V
54elpr 3993 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
65bibi2i 313 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
76albii 1611 . . . . 5  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  <->  A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
83, 7bitri 249 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
( w  =  x  \/  w  =  y ) ) )
98exbii 1635 . . 3  |-  ( E. z  z  =  {
x ,  y }  <->  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
102, 9mpbir 209 . 2  |-  E. z 
z  =  { x ,  y }
1110issetri 3075 1  |-  { x ,  y }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3068   {cpr 3977
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 4511  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-v 3070  df-un 3431  df-sn 3976  df-pr 3978
This theorem is referenced by:  snex  4631  prex  4632  pwssun  4725  xpsspwOLD  5052  funopg  5548  fiint  7689  brdom7disj  8799  brdom6disj  8800  wlkntrllem1  23593  frisusgranb  30727  2pthfrgrarn  30739
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