Theorem inidm 3316

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 394	. 2 |- ( ( x e. A /\ x e. A ) <-> x e. A )
2	1	ineqri 3300	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1335   e. wcel 2128   i^i cin 3101

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108

This theorem is referenced by:  inindi  3324  inindir  3325  uneqin  3358  ssdifeq0  3476  intsng  3841  xpindi  4718  xpindir  4719  resindm  4905  ofres  6040  offval2  6041  ofrfval2  6042  suppssof1  6043  ofco  6044  offveqb  6045  caofref  6047  caofrss  6050  caoftrn  6051  undifdc  6861  baspartn  12408  epttop  12450  dvaddxxbr  13025  dvmulxxbr  13026  dvaddxx  13027  dvmulxx  13028  dviaddf  13029  dvimulf  13030