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Theorem bj-sels 10034
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 3400 . . 3  |-  ( A  e.  V  ->  A  e.  { A } )
2 bj-snexg 10032 . . . . 5  |-  ( A  e.  V  ->  { A }  e.  _V )
3 sbcel2g 2871 . . . . 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 2877 . . . . . 6  |-  ( { A }  e.  _V  ->  [_ { A }  /  x ]_ x  =  { A } )
62, 5syl 14 . . . . 5  |-  ( A  e.  V  ->  [_ { A }  /  x ]_ x  =  { A } )
76eleq2d 2107 . . . 4  |-  ( A  e.  V  ->  ( A  e.  [_ { A }  /  x ]_ x  <->  A  e.  { A }
) )
84, 7bitrd 177 . . 3  |-  ( A  e.  V  ->  ( [. { A }  /  x ]. A  e.  x  <->  A  e.  { A }
) )
91, 8mpbird 156 . 2  |-  ( A  e.  V  ->  [. { A }  /  x ]. A  e.  x
)
109spesbcd 2844 1  |-  ( A  e.  V  ->  E. x  A  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   _Vcvv 2557   [.wsbc 2764   [_csb 2852   {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-pr 3944  ax-bdor 9936  ax-bdeq 9940  ax-bdsep 10004
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382
This theorem is referenced by: (None)
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