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Theorem bj-zfpair2 11458
Description: Proof of zfpair2 4028 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 11368 . . . . 5  |- BOUNDED  w  =  x
2 ax-bdeq 11368 . . . . 5  |- BOUNDED  w  =  y
31, 2ax-bdor 11364 . . . 4  |- BOUNDED  ( w  =  x  \/  w  =  y )
4 ax-pr 4027 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
53, 4bdbm1.3ii 11439 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
6 dfcleq 2082 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
7 vex 2622 . . . . . . . 8  |-  w  e. 
_V
87elpr 3462 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
98bibi2i 225 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
109albii 1404 . . . . 5  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  <->  A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
116, 10bitri 182 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
( w  =  x  \/  w  =  y ) ) )
1211exbii 1541 . . 3  |-  ( E. z  z  =  {
x ,  y }  <->  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
135, 12mpbir 144 . 2  |-  E. z 
z  =  { x ,  y }
1413issetri 2628 1  |-  { x ,  y }  e.  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 103    \/ wo 664   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-pr 4027  ax-bdor 11364  ax-bdeq 11368  ax-bdsep 11432
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by:  bj-prexg  11459
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