ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inidm Unicode version

Theorem inidm 3253
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 391 . 2  |-  ( ( x  e.  A  /\  x  e.  A )  <->  x  e.  A )
21ineqri 3237 1  |-  ( A  i^i  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463    i^i cin 3038
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045
This theorem is referenced by:  inindi  3261  inindir  3262  uneqin  3295  ssdifeq0  3413  intsng  3773  xpindi  4642  xpindir  4643  resindm  4829  ofres  5962  offval2  5963  ofrfval2  5964  suppssof1  5965  ofco  5966  offveqb  5967  caofref  5969  caofrss  5972  caoftrn  5973  undifdc  6778  baspartn  12123  epttop  12165  dvaddxxbr  12740  dvmulxxbr  12741  dvaddxx  12742  dvmulxx  12743  dviaddf  12744  dvimulf  12745
  Copyright terms: Public domain W3C validator