Theorem inidm 3280

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 393	. 2 |- ( ( x e. A /\ x e. A ) <-> x e. A )
2	1	ineqri 3264	1 |- ( A i^i A ) = A

Syntax hints:   = wceq 1331   e. wcel 1480   i^i cin 3065

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072

This theorem is referenced by:  inindi  3288  inindir  3289  uneqin  3322  ssdifeq0  3440  intsng  3800  xpindi  4669  xpindir  4670  resindm  4856  ofres  5989  offval2  5990  ofrfval2  5991  suppssof1  5992  ofco  5993  offveqb  5994  caofref  5996  caofrss  5999  caoftrn  6000  undifdc  6805  baspartn  12206  epttop  12248  dvaddxxbr  12823  dvmulxxbr  12824  dvaddxx  12825  dvmulxx  12826  dviaddf  12827  dvimulf  12828