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Theorem bj-zfpair2 14598
Description: Proof of zfpair2 4210 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 14508 . . . . 5  |- BOUNDED  w  =  x
2 ax-bdeq 14508 . . . . 5  |- BOUNDED  w  =  y
31, 2ax-bdor 14504 . . . 4  |- BOUNDED  ( w  =  x  \/  w  =  y )
4 ax-pr 4209 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
53, 4bdbm1.3ii 14579 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
6 dfcleq 2171 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
7 vex 2740 . . . . . . . 8  |-  w  e. 
_V
87elpr 3613 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
98bibi2i 227 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
109albii 1470 . . . . 5  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  <->  A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
116, 10bitri 184 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
( w  =  x  \/  w  =  y ) ) )
1211exbii 1605 . . 3  |-  ( E. z  z  =  {
x ,  y }  <->  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
135, 12mpbir 146 . 2  |-  E. z 
z  =  { x ,  y }
1413issetri 2746 1  |-  { x ,  y }  e.  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 105    \/ wo 708   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   _Vcvv 2737   {cpr 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-pr 4209  ax-bdor 14504  ax-bdeq 14508  ax-bdsep 14572
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599
This theorem is referenced by:  bj-prexg  14599
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