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Theorem bj-prexg 13253
Description: Proof of prexg 4136 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 3604 . . . . . 6  |-  ( y  =  B  ->  { x ,  y }  =  { x ,  B } )
21eleq1d 2208 . . . . 5  |-  ( y  =  B  ->  ( { x ,  y }  e.  _V  <->  { x ,  B }  e.  _V ) )
3 bj-zfpair2 13252 . . . . 5  |-  { x ,  y }  e.  _V
42, 3vtoclg 2746 . . . 4  |-  ( B  e.  W  ->  { x ,  B }  e.  _V )
5 preq1 3603 . . . . 5  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
65eleq1d 2208 . . . 4  |-  ( x  =  A  ->  ( { x ,  B }  e.  _V  <->  { A ,  B }  e.  _V ) )
74, 6syl5ib 153 . . 3  |-  ( x  =  A  ->  ( B  e.  W  ->  { A ,  B }  e.  _V ) )
87vtocleg 2757 . 2  |-  ( A  e.  V  ->  ( B  e.  W  ->  { A ,  B }  e.  _V ) )
98imp 123 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686   {cpr 3528
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-pr 4134  ax-bdor 13158  ax-bdeq 13162  ax-bdsep 13226
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534
This theorem is referenced by:  bj-snexg  13254  bj-unex  13261
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