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Theorem elin 3123
Description: Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
Assertion
Ref Expression
elin  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )

Proof of Theorem elin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2563 . 2  |-  ( A  e.  ( B  i^i  C )  ->  A  e.  _V )
2 elex 2563 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
32adantl 262 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A  e.  _V )
4 eleq1 2100 . . . 4  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
5 eleq1 2100 . . . 4  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
64, 5anbi12d 442 . . 3  |-  ( x  =  A  ->  (
( x  e.  B  /\  x  e.  C
)  <->  ( A  e.  B  /\  A  e.  C ) ) )
7 df-in 2921 . . 3  |-  ( B  i^i  C )  =  { x  |  ( x  e.  B  /\  x  e.  C ) }
86, 7elab2g 2686 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) ) )
91, 3, 8pm5.21nii 620 1  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   _Vcvv 2554    i^i cin 2913
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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
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-v 2556  df-in 2921
This theorem is referenced by:  elin2  3124  elin3  3125  incom  3126  ineqri  3127  ineq1  3128  inass  3144  inss1  3154  ssin  3156  ssrin  3159  inssdif  3170  difin  3171  unssin  3173  inssun  3174  invdif  3176  indif  3177  indi  3181  undi  3182  difundi  3186  difindiss  3188  indifdir  3190  difin2  3196  inrab2  3207  inelcm  3279  inssdif0im  3288  uniin  3597  intun  3643  intpr  3644  elrint  3652  iunin2  3717  iinin2m  3722  elriin  3724  brin  3808  trin  3861  inex1  3888  inuni  3906  bnd2  3923  ordpwsucss  4260  ordpwsucexmid  4263  peano5  4283  inopab  4430  inxp  4432  dmin  4505  opelres  4579  intasym  4671  asymref  4672  dminss  4700  imainss  4701  inimasn  4703  ssrnres  4725  cnvresima  4772  dfco2a  4783  imainlem  4942  imain  4943  2elresin  4972  nfvres  5168  respreima  5257  isoini  5419  offval  5681  tfrlem5  5892  peano5nnnn  6921  peano5nni  7867  ixxdisj  8713  icodisj  8801  fzdisj  8857  uzdisj  8896  nn0disj  8936  fzouzdisj  8977  bdinex1  9866  bj-indind  9903  peano5setOLD  9912
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