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Theorem inidm 3176
Description: Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
inidm  |-  ( A  i^i  A )  =  A

Proof of Theorem inidm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 anidm 388 . 2  |-  ( ( x  e.  A  /\  x  e.  A )  <->  x  e.  A )
21ineqri 3160 1  |-  ( A  i^i  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434    i^i cin 2973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980
This theorem is referenced by:  inindi  3184  inindir  3185  uneqin  3216  ssdifeq0  3326  intsng  3672  xpindi  4493  xpindir  4494  resindm  4674  ofres  5750  offval2  5751  ofrfval2  5752  suppssof1  5753  ofco  5754  offveqb  5755  caofref  5757  caofrss  5760  caoftrn  5761  undiffi  6433
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