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Theorem bj-sels 12935
Description: If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.)
Assertion
Ref Expression
bj-sels  |-  ( A  e.  V  ->  E. x  A  e.  x )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem bj-sels
StepHypRef Expression
1 snidg 3522 . . 3  |-  ( A  e.  V  ->  A  e.  { A } )
2 bj-snexg 12933 . . . . 5  |-  ( A  e.  V  ->  { A }  e.  _V )
3 sbcel2g 2992 . . . . 5  |-  ( { A }  e.  _V  ->  ( [. { A }  /  x ]. A  e.  x  <->  A  e.  [_ { A }  /  x ]_ x ) )
42, 3syl 14 . . . 4  |-  ( A  e.  V  ->  ( [. { A }  /  x ]. A  e.  x  <->  A  e.  [_ { A }  /  x ]_ x
) )
5 csbvarg 2998 . . . . . 6  |-  ( { A }  e.  _V  ->  [_ { A }  /  x ]_ x  =  { A } )
62, 5syl 14 . . . . 5  |-  ( A  e.  V  ->  [_ { A }  /  x ]_ x  =  { A } )
76eleq2d 2185 . . . 4  |-  ( A  e.  V  ->  ( A  e.  [_ { A }  /  x ]_ x  <->  A  e.  { A }
) )
84, 7bitrd 187 . . 3  |-  ( A  e.  V  ->  ( [. { A }  /  x ]. A  e.  x  <->  A  e.  { A }
) )
91, 8mpbird 166 . 2  |-  ( A  e.  V  ->  [. { A }  /  x ]. A  e.  x
)
109spesbcd 2965 1  |-  ( A  e.  V  ->  E. x  A  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   _Vcvv 2658   [.wsbc 2880   [_csb 2973   {csn 3495
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-pr 4099  ax-bdor 12837  ax-bdeq 12841  ax-bdsep 12905
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-sn 3501  df-pr 3502
This theorem is referenced by: (None)
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