Theorem inidm 3209

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.

Step	Hyp	Ref	Expression
1		anidm 388	. 2 |- ( ( x e. A /\ x e. A ) <-> x e. A )
2	1	ineqri 3193	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1289   e. wcel 1438   i^i cin 2998

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005

This theorem is referenced by:  inindi  3217  inindir  3218  uneqin  3250  ssdifeq0  3365  intsng  3722  xpindi  4571  xpindir  4572  resindm  4754  ofres  5869  offval2  5870  ofrfval2  5871  suppssof1  5872  ofco  5873  offveqb  5874  caofref  5876  caofrss  5879  caoftrn  5880  undifdc  6634