Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-sels Unicode version

Theorem bj-sels 14548
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 3621 . . 3  |-  ( A  e.  V  ->  A  e.  { A } )
2 bj-snexg 14546 . . . . 5  |-  ( A  e.  V  ->  { A }  e.  _V )
3 sbcel2g 3078 . . . . 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 3085 . . . . . 6  |-  ( { A }  e.  _V  ->  [_ { A }  /  x ]_ x  =  { A } )
62, 5syl 14 . . . . 5  |-  ( A  e.  V  ->  [_ { A }  /  x ]_ x  =  { A } )
76eleq2d 2247 . . . 4  |-  ( A  e.  V  ->  ( A  e.  [_ { A }  /  x ]_ x  <->  A  e.  { A }
) )
84, 7bitrd 188 . . 3  |-  ( A  e.  V  ->  ( [. { A }  /  x ]. A  e.  x  <->  A  e.  { A }
) )
91, 8mpbird 167 . 2  |-  ( A  e.  V  ->  [. { A }  /  x ]. A  e.  x
)
109spesbcd 3049 1  |-  ( A  e.  V  ->  E. x  A  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   _Vcvv 2737   [.wsbc 2962   [_csb 3057   {csn 3592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-pr 4209  ax-bdor 14450  ax-bdeq 14454  ax-bdsep 14518
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-sn 3598  df-pr 3599
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator