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Theorem bj-zfpair2 13035
Description: Proof of zfpair2 4102 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-zfpair2  |-  { x ,  y }  e.  _V

Proof of Theorem bj-zfpair2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-bdeq 12945 . . . . 5  |- BOUNDED  w  =  x
2 ax-bdeq 12945 . . . . 5  |- BOUNDED  w  =  y
31, 2ax-bdor 12941 . . . 4  |- BOUNDED  ( w  =  x  \/  w  =  y )
4 ax-pr 4101 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
53, 4bdbm1.3ii 13016 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
6 dfcleq 2111 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
7 vex 2663 . . . . . . . 8  |-  w  e. 
_V
87elpr 3518 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
98bibi2i 226 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
109albii 1431 . . . . 5  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  <->  A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
116, 10bitri 183 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
( w  =  x  \/  w  =  y ) ) )
1211exbii 1569 . . 3  |-  ( E. z  z  =  {
x ,  y }  <->  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
135, 12mpbir 145 . 2  |-  E. z 
z  =  { x ,  y }
1413issetri 2669 1  |-  { x ,  y }  e.  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 682   A.wal 1314    = wceq 1316   E.wex 1453    e. wcel 1465   _Vcvv 2660   {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-pr 4101  ax-bdor 12941  ax-bdeq 12945  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  bj-prexg  13036
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