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Theorem zfpair2 4640
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4639. See zfpair 4637 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 4639 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
21bm1.3ii 4528 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
3 dfcleq 2445 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 3048 . . . . . . . 8  |-  w  e. 
_V
54elpr 3986 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
65bibi2i 315 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
76albii 1691 . . . . 5  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  <->  A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
83, 7bitri 253 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
( w  =  x  \/  w  =  y ) ) )
98exbii 1718 . . 3  |-  ( E. z  z  =  {
x ,  y }  <->  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
102, 9mpbir 213 . 2  |-  E. z 
z  =  { x ,  y }
1110issetri 3052 1  |-  { x ,  y }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    \/ wo 370   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045   {cpr 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-un 3409  df-sn 3969  df-pr 3971
This theorem is referenced by:  snex  4641  prex  4642  pwssun  4740  xpsspw  4948  funopg  5614  fiint  7848  brdom7disj  8959  brdom6disj  8960  wlkntrllem1  25289  frisusgranb  25725  2pthfrgrarn  25737
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