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Theorem bj-prexg 11459
Description: Proof of prexg 4029 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-prexg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )

Proof of Theorem bj-prexg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3515 . . . . . 6  |-  ( y  =  B  ->  { x ,  y }  =  { x ,  B } )
21eleq1d 2156 . . . . 5  |-  ( y  =  B  ->  ( { x ,  y }  e.  _V  <->  { x ,  B }  e.  _V ) )
3 bj-zfpair2 11458 . . . . 5  |-  { x ,  y }  e.  _V
42, 3vtoclg 2679 . . . 4  |-  ( B  e.  W  ->  { x ,  B }  e.  _V )
5 preq1 3514 . . . . 5  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
65eleq1d 2156 . . . 4  |-  ( x  =  A  ->  ( { x ,  B }  e.  _V  <->  { A ,  B }  e.  _V ) )
74, 6syl5ib 152 . . 3  |-  ( x  =  A  ->  ( B  e.  W  ->  { A ,  B }  e.  _V ) )
87vtocleg 2690 . 2  |-  ( A  e.  V  ->  ( B  e.  W  ->  { A ,  B }  e.  _V ) )
98imp 122 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    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-snexg  11460  bj-unex  11467
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