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Theorem bj-zfpair2 10417
Description: Proof of zfpair2 3973 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 10327 . . . . 5  |- BOUNDED  w  =  x
2 ax-bdeq 10327 . . . . 5  |- BOUNDED  w  =  y
31, 2ax-bdor 10323 . . . 4  |- BOUNDED  ( w  =  x  \/  w  =  y )
4 ax-pr 3972 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
53, 4bdbm1.3ii 10398 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
6 dfcleq 2050 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
7 vex 2577 . . . . . . . 8  |-  w  e. 
_V
87elpr 3424 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
98bibi2i 220 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
109albii 1375 . . . . 5  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  <->  A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
116, 10bitri 177 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
( w  =  x  \/  w  =  y ) ) )
1211exbii 1512 . . 3  |-  ( E. z  z  =  {
x ,  y }  <->  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
135, 12mpbir 138 . 2  |-  E. z 
z  =  { x ,  y }
1413issetri 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-pr 3972  ax-bdor 10323  ax-bdeq 10327  ax-bdsep 10391
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:  bj-prexg  10418
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