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Theorem dfin5 3160
Description: Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
Assertion
Ref Expression
dfin5  |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
Distinct variable groups:    x, A    x, B

Proof of Theorem dfin5
StepHypRef Expression
1 df-in 3159 . 2  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
2 df-rab 2552 . 2  |-  { x  e.  A  |  x  e.  B }  =  {
x  |  ( x  e.  A  /\  x  e.  B ) }
31, 2eqtr4i 2306 1  |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547    i^i cin 3151
This theorem is referenced by:  nfin  3375  rabbi2dva  3377  dfepfr  4378  epfrc  4379  ablfaclem3  15322  mretopd  16829  ptclsg  17309  xkopt  17349  iscmet3  18719  xrlimcnp  20263  ppiub  20443  suppss2f  23201  xppreima  23211  orvcelval  23669  sstotbnd2  26498  pmtrmvd  27398  glbconN  29566  2polssN  30104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-cleq 2276  df-rab 2552  df-in 3159
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