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Theorem bj-sels 11760
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 3473 . . 3  |-  ( A  e.  V  ->  A  e.  { A } )
2 bj-snexg 11758 . . . . 5  |-  ( A  e.  V  ->  { A }  e.  _V )
3 sbcel2g 2952 . . . . 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 2958 . . . . . 6  |-  ( { A }  e.  _V  ->  [_ { A }  /  x ]_ x  =  { A } )
62, 5syl 14 . . . . 5  |-  ( A  e.  V  ->  [_ { A }  /  x ]_ x  =  { A } )
76eleq2d 2157 . . . 4  |-  ( A  e.  V  ->  ( A  e.  [_ { A }  /  x ]_ x  <->  A  e.  { A }
) )
84, 7bitrd 186 . . 3  |-  ( A  e.  V  ->  ( [. { A }  /  x ]. A  e.  x  <->  A  e.  { A }
) )
91, 8mpbird 165 . 2  |-  ( A  e.  V  ->  [. { A }  /  x ]. A  e.  x
)
109spesbcd 2925 1  |-  ( A  e.  V  ->  E. x  A  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   [.wsbc 2840   [_csb 2933   {csn 3446
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 4036  ax-bdor 11662  ax-bdeq 11666  ax-bdsep 11730
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-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-sn 3452  df-pr 3453
This theorem is referenced by: (None)
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