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

Theorem bj-sels 13949
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 3612 . . 3  |-  ( A  e.  V  ->  A  e.  { A } )
2 bj-snexg 13947 . . . . 5  |-  ( A  e.  V  ->  { A }  e.  _V )
3 sbcel2g 3070 . . . . 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 3077 . . . . . 6  |-  ( { A }  e.  _V  ->  [_ { A }  /  x ]_ x  =  { A } )
62, 5syl 14 . . . . 5  |-  ( A  e.  V  ->  [_ { A }  /  x ]_ x  =  { A } )
76eleq2d 2240 . . . 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 3041 1  |-  ( A  e.  V  ->  E. x  A  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   [.wsbc 2955   [_csb 3049   {csn 3583
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-pr 4194  ax-bdor 13851  ax-bdeq 13855  ax-bdsep 13919
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator