ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  zfpair2 Unicode version

Theorem zfpair2 3973
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 3972. (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 3972 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
21bm1.3ii 3906 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
3 dfcleq 2050 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 2577 . . . . . . . 8  |-  w  e. 
_V
54elpr 3424 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
65bibi2i 220 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
76albii 1375 . . . . 5  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  <->  A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
83, 7bitri 177 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
( w  =  x  \/  w  =  y ) ) )
98exbii 1512 . . 3  |-  ( E. z  z  =  {
x ,  y }  <->  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
102, 9mpbir 138 . 2  |-  E. z 
z  =  { x ,  y }
1110issetri 2581 1  |-  { x ,  y }  e.  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 102    \/ wo 639   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574   {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410
This theorem is referenced by:  prexgOLD  3974  prexg  3975  onintexmid  4325  funopg  4962
  Copyright terms: Public domain W3C validator