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Theorem zfpair2 4687
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4686. See zfpair 4684 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 4686 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
21bm1.3ii 4571 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
3 dfcleq 2460 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 3116 . . . . . . . 8  |-  w  e. 
_V
54elpr 4045 . . . . . . 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 1620 . . . . 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 1644 . . 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 3120 1  |-  { x ,  y }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-sn 4028  df-pr 4030
This theorem is referenced by:  snex  4688  prex  4689  pwssun  4786  xpsspwOLD  5115  funopg  5618  fiint  7793  brdom7disj  8905  brdom6disj  8906  wlkntrllem1  24237  frisusgranb  24673  2pthfrgrarn  24685
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